straigt lines (2)

8/11/2019 Straigt Lines (2)

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MATHEMATICSBY

Attri D.****** Mathematics by Attri. D. ******

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STRAIGHT LINES

****** Mathematics by Attri. D. ******

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nzks.kkpk;Z Academy ResonantClassesS.C. F. 57, Sector 7 S.C.O. 53, Sector 17

Kurukshetra Kurukshetra

STRAIGHT LINES

IMPORTANT TOOLS/ TECHNIQUES / TIPS

1. DIFFERENT FORMS OF STRAIGHT LINES EQUATION

(i) If a straight line is p arallel to the axis of x a nd is a t a distance b

from the origin, its equation is y = b. In particular, equation of x-

axis is y = 0.

(ii) The equation of a straight line parallel to the axis of y at a

distance a from origin is x = a. In particular equation of y -axis is

x = 0.

(iii) The equation of a line passing through ( )11 , y x and having theslope m is ( )11 x xm y y = .

(iv)Straight line cutting intercepts a and b on x and y-axis respectively

is 1=+b y

a x

.

(v)Straight line having the slope m and cutting the intercept c o n y-

axis i s cmx y += .(vi)Straight line w hich is at a distance p from the origin and the angle

which the perpendicular makes with x-axis is , is p y x =+ sincos .

(vii)If a line makes an angle with positive x-axis, and passes

through ( )11 , y x then its equation is r y y x x == sincos 11 , where r isthe directed distance of any point ( ) y xm , from the point ( )11 , y x .

Any point on this line will be sin,cos 11 r y yr x x +=+= .2. Perpendicular distance of a point ( )11 , y x from the line 0=++ cby Ax

is22

11

B A

C By Ax

+++

, in particular the perpendicular distance of origin

from the line 0=++ C By Ax must be 22 B AC

+= .


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3. The distance between two parallel lines 0=++ C By Ax ,0=++ C By Ax is equal to difference of perpendicular distances of them

from the origin, i.e.,22 B A

C C

+

.

4. The equation of family of lines passing through the intersection oftwo lines 0=++ C By Ax and 0=++ C y B x A is

( ) 0=+++++ C y B x Ak C By Ax , where k is a parameter.

5. Let L = 0 be the equation of a line we dene:

(i) ( )11 , y x L as n umerical value of equation of L if we su bstitute theco-ordinates of a point ( )11 , y x in L (i.e., 11 , y y x x == ).

(ii) ( ) P L as the numerical value of equation of L, if we put the co-

ordinates of a point P in the e quation of L.Now if ( ) ( ) 0>Q L P L the points P and Q lie on the same side of L = 0and if ( ) ( ) 0C C then by taking + sign in the

above equation, we get the equation of angular bisector bisecting theangle containing origin.

8. (i)If three lines 0=++ iii C y B x A , ( )3,2,1=i are concurrent thenconversely, if D = 0 they are either con current or p arallel.

(ii) Three pair wise unparallel lines 0,0 21 == L L and 03 = L will be

concurrent or p arallel if there exists numbers p, q, r n ot a ll zerosuch that 0321 =++ rLqL pL .

9. Change of Origin. Let Ox, Oy be the original axes and the origin Ois taken to a point (h, k) with reference to O such that axes remain

parallel to original axes. Let P be a point whose co-ordinates withreference to O are (x, y) and with reference to O are (X, Y) then

k Y yh X x +=+= , . Note that by this transformation slope of a line

remains same. ConsequentlydX dY

dxdy = .


3

y

P ( X, Y )

0

(h, k )

x

Y

0 X


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10. Rotation of Axes. Let the axes Ox and Oy be rotated through an

angle in the counter clockwise direction such that the new axes

become xO and yO . Then if P has co-ordinates (x, y) with referenceto original axes and has co-ordinates ( ) y x , with reference to newaxes, we have sincos y x x = and cossin y x y += .

Also cossin,sincos y x y y x x +=+= .

11. Locus. Finding the locus of a point ( ) , is e quivalent to nding arelation between and free from arbitrary constants (parameters).In case the point whose locus is to be found happens to be a point of

intersection of two lines (curves), the locus is simply the eliminant of

their equ ations. Note that for ea ch real t, we get a point in the planewhere 2,12 =+= t yt x . But for a ll t the relation (which is obtained by

eliminating t) ( ) 12 2 ++= y x is satised. Therefore the locus of a moving point whose abscissa and ordinate are functions of a real parameter is

obtained by eliminating the parameter and getting a pure relation

between x and y.

12. The general equation of second degree0222 22 =+++++= c fy gxbyhxyaxG may represents a pair of lines if


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0=

c f g

f bh

g ha or 02 222 =+ chbg af fghabc which is p recisely the condition

under which G can be written as a product of two linear factors not

necessarily real. These lines will be real if abh 2 .13. The Homogeneous equation of second degree 02 22 =++= byhxyax H

always represents a pair of lines passing through origin if abh 2 .

The angle between these lines is given by,ba

abh

+

=22

tan these

lines will be perpendicular if a + b = 0.

14. Let ( ) 0, = y x f be a second degree curve which is intersected by aline at A and B. Then combined equation of OA and OB can be

obtained by making ( ) 0, = y x f homogeneous w ith the help of the linesequation.

15. The equation of lines bisecting the angles between the lines

02 22 =++ byhxyax ish

xyba y x =

22

.

16. Let ( ) 0, = y x g be a second degree curve which is intersected by aline at A and B. Then the equation of pair of lines OA and OB can be

obtained by making ( ) 0, = y x g homogeneous with the help of thelines equation.

STANDARD RESULTS AND IMPORTANT TIPS

(i) If ( ) ( ) ( )332211 ,,,,, y xC y x B y x A be the vertices of a triangle ABC thenco-ordinates of various centre are given bycentre Co-ordinates

Centroid

++++

3,

3321321 y y y x x x


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Incentre

++++

++++

cbacybyay

cbacxbxax 321321 ,

Orthocentre

++++++++

C B A

C y B y A y

C B AC x B x A x

tantantan

tantantan

tantantan

tantantan

321

321

Circumcentre

++++

++++

C B A

C y B y A y

C B AC x B x A x

2sin2sin2sin

2sin2sin2sin

2sin2sin2sin

2sin2sin2sin

321

321

(ii) Two lines 0111 =++ C y B x A and 0222 =++ C y B x A are coincident

2

1

2

1

2

1

C

C

B

B

A

A == if2

1

2

1

B

B

A

A = but if they are not equal to2

1

C

C then the lines

are parallel b ut n ot coi ncident. If2

1

2

1

B

B

A

A or 02211 B A B A then the

lines are intersecting.

(iii) Let ABC be a triangle and M be a point. The point M will lie with in

triangle ABC if and only if

(a) M and BC lie on the same side of A.

(b) M and AC lie on the same side of B.

(c) M and AB lie on the same side of C.

(iv) The point of intersection of pair of straight lines

( ) 0222, 22 =+++++= c fy gxhxybyax y x f can be obtained by differentiating( ) y x f , partially differentiating w.r.t. x a nd y and then solve for x a nd

y i.e., 0=++ g hyax , 0=++ f byhx . 21

habaf hg

f

bg hf

x

=

=

,

22 , habaf hg y

habbg hf x

=

= .


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(v) Area of formed by the three lines 2211 , c xm yc xm y +=+= and

33 cm y += is( )

=

21

221

21

mmcc

A .

(vi) Area of the triangle formed by the line 0=++ cbyax a nd the co-ordinate

axis is abc

2

2

.

(vii) Area of rhombus formed by the lines 0= cbyax is abc 22

.

(viii) Area of parallelogram, the equation of whose side a re, 0111 =++ c yb xa ,

0111 =++ d yb xa , 0222 =++ c yb xa , 0222 =++ d yb xa is( ) ( )

1221

2221

babacd cd

.

(ix) The co-ordinate of the foot of perpendicular (h, k) from ( )11 , y x to theline 0=++ cbyax is

( )22

1111

bacbyax

bk y

ah x

+

++=

=

.

Also the image of the point ( )11 , y x is g iven by( )

22

1111 2

ba

cbyax

b

k y

a

h x

+++== .


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Type 1 (NCERT QUESTIONS)E X E R C I E 1 0 . 1

E.1. Find the slope of the lines :

(a) Passing through the points (3, 2) and (1, 4)

(b) Passing through the points (3, 2) and (7, 2)

(c) Passing through the points (3, 2) and (3,4)

(d) Making inclination of 60 0 with the positive direction of

x-axis.

E.2. If the angle between two line is4

and slope of one of the lines

is 21

, nd the slope of the other line.

E.3. Line through the points (2,6) and (4, 8) is perpendicular to the

line through the points (8,12) and (x, 24). Find the value of x.

E.4. Three points P (h,k), Q(x 1 , y 1 ) and R (x 2 , y 2 ) lie on a line. Show

that (h x 1 ) (y 2 y 1 ) = (k y 1 ) (x 2 x 1 ).

E.5. In g., time and distance graph of a

linear motion is given. Two positions

of time and distance are recorded as,

when T = 0, D =2 and when T =3,

D = 8. Using the concept of slope,

nd law of motion, i.e., how distance

depends upon time.

1. Draw a quadrilateral in the Cartesian plane, whose vertices are

( 4 , 5), (0, 7), (5, 5 ) and ( 4 , 2). Also, nd its area.

2. The base of an equilateral triangle with side 2a lies along the y-axissuch that the mid-point of the base is at the origin. Find vertices of t he

triangle.

3. Find the distance between P (x 1, y 1) and Q (x 2 , y 2) when :

(i) PQ is parallel to the y-axis, (ii) PQ is parallel to the x-axis.

4. Find a point on the x-axis, which is equidistant from the points (7, 6)

and (3, 4).

5. Find the slope of a line, which passes t hrough the origin, and the mid-

point of the line segment joining the point


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( P o p u l a t i o n

i n

c r o r e s )


6. Without using the Pythagoras theorem, show that the points (4, 4),

(3, 5) and (1, 1) are the vertices of a right angled triangle.

7. Find the slope of the line, which makes a n angle of 30 with the positive

direction of y-axis measured anticlockwise.

8. Find the value of x for which the points (x,1), (2,1) and (4, 5) are

collinear.

9. Without using distance formula, show that points (2, 1), (4, 0), (3, 3)

and (3, 2) are the vertices of a parallelogram.

10. Find the angle between the x-axis and the line joining the points (3,1)

and (4,2).

11. The s lope of a line is d ouble of the s lope o f another line. If tangent of the

angle b etween them is31 , nd the slopes of the lines.

12. A line passes through (x 1, y 1) and (h, k). If slope of t he line is m, show

that k y 1 = m (h x 1).

13. If three points ( h, 0), (a, b) and (0, k) lie o n a line, show that 1=+k b

ha

14. Consider the following population and year graph g. nd the slope of

the line AB and using it, nd what will be the population in the year

2010?

E X E R C I E 1 0 . 2

E.6. Find the equation of the lines parallel to axes and passing through

(2,3).

E.7. Find the e quation of the l ine t hrough (2,3) with slope 4 .

E.8. Write t he e quation of the l ine t hrough the p oints (1, 1) and (3,5)


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E.9. Write the equation of the lines for which tan2

1= , where is the

inclination of the line and (i) y i ntercept is2

3 (ii) x intercepts is 4.

E.10.Find the equation of the line, which makes intercepts 3 and 2 on thex a nd y a xes respectively.

E.11.Find the equation of the line whose perpendicular d istance from the

origin is 4 units and the angle which the normal makes with positive

direction of x-axis is 15.

E.12.The Fahrenheit temperature F and absolute temperature K satisfy a

linear equation. Given that K = 273 when F = 32 and that K = 373

when F = 2 12. Express K in terms of F and nd the value of F, when K

= 0.

In Exercises 1 to 8, nd the equation of the line which satisfy the given

conditions:

1. Write the equations for the x-and y-axes.

2. Passing through the point ( 4, 3) with slope21

3. Passing through (0, 0) with slope m.

4. Passing through ( 32,2 ) and inclined with the x -axis a t an angle of 75 0 .

5. Intersecting the x-axis at a distance o f 3 units to the left of origin with

slope 2 .

6. Intersecting the y-axis at a distance of 2 units above the origin and

making an angle of 30 0 w ith positive direction of the x-axis.

7. Passing through the points (1, 1) and (2, 4).

8. Perpendicular distance from the origin is 5 units a nd the angle made by

the perpendicular with the positive x-axis is 30 0 .

9. The vertices of PQR are P (2, 1), Q (2, 3) and R (4, 5). Find equation ofthe m edian through the ver tex R.

10. Find the equ ation of the line passing through (3, 5) and perpendicular

to the line through the points (2, 5) and (3, 6).

11. A line perpendicular to the line segment joining the points (1, 0) and

(2, 3) divides it in the ratio 1: n. Find the equation of t he line.

12. Find the equation of a line that cuts off equal intercepts on the

coordinate axes and passes t hrough the p oint (2, 3).


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13. Find equation of the line p assing through the p oint (2, 2) and cutting off

intercepts on the a xes w hose su m is 9 .

14. Find equation of the line through the point (0, 2) making an angle3

2

with the positive x-axis. Also, nd the equation of line p arallel to it andcrossing the y-axis a t a distance o f 2 units b elow the origin.

15. The perpendicular from the origin to a line m eets it at the point (2, 9),

nd the equation of the line.

16. The length L (in centimetrs) of a copper rod is a linear function of its

Celsius temperature C. In an experiment, if L = 124.942 when C = 20

and L= 125.134 when C = 110, express L i n terms of C.

17. The owner of a milk store nds that, he ca n sell 980 litres of milk each

week at Rs 14/litre and 1220 litres of milk each week at Rs 16/litre.

Assuming a linear relationship between selling price and emand, how

many litres could he s ell weekly at Rs 1 7/litre?

18. P (a, b) is the mid-point of a line segm ent between axes. Show that

equation of the line is 2=+b y

a x

.

19. Point R (h, k) divides a line segm ent between the a xes in the ratio 1: 2.

Find equation of the l ine.

20. By using the concept of equation of a line, prove that the three points

(3, 0), ( 2, 2) and (8, 2) are col linear.

E x e r c i s e 1 0 . 3

E.13. Equation of a line is 3 x 4 y + 10 = 0. Find its

(i) slope, (ii) x and y-intercepts.

E. 14.Reduce the equation 083 =+ y x into normal form. Find the va lues of

P and .

E. 15.Find the a ngle between the lines 063053 =+= x yand x y .E. 16.Show that two lines 0,00 21222111 =++=++ bbwherec yb xaand c yb xa

are: (i) Parallel if2

2

1

1

b

a

b

a = , and (ii) Perpendicular if 02121 =+ bbaa .

E. 17.Find the equation of a line perpendicular to the line x 2y + 3 =0 and

passing through the point (1, 2).

E. 18.Find the d istance of the p oint (3, 5) from the line 3 x 4 y 26 =0.

E. 19.Find the distance between the parallel lines 3x 4y +7 =0 and

3x 4y +5 =0


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1. Reduce t he following equations into s lope - intercept form and nd their

slopes and the y - intercepts.

(i) x + 7y = 0, (ii) 6x + 3y 5 = 0, (iii) y = 0.

2. Reduce the following equations into intercept form and nd their

intercepts o n the a xes.

(i) 3x + 2y 12 = 0, (ii) 4x 3y = 6, (iii) 3y + 2 = 0.

3. Reduce the following equations into normal form. Find their

perpendicular distances from the origin and angle between

perpendicular and the positive x-axis.

(i) x 3 y + 8 = 0, (ii) y 2 = 0, (iii) x y = 4.

4. Find the d istance of the p oint (1, 1) from the line 1 2(x + 6) = 5(y 2).

5. Find the points on the x-axis, whose distances from the line 143 =+ y x

are 4 units.

6. Find the distance b etween p arallel lines

(i) 15x + 8 y 34 = 0 and 15x + 8 y + 31 = 0

(ii) l (x + y) + p = 0 and l (x + y) r = 0.

7. Find equation of the line pa rallel to the line 3x 4y + 2 = 0 and passing

through the point (2, 3).

8. Find equation of the line perpendicular to the line x 7y + 5 = 0 andhaving x intercept 3.

9. Find angles between the lines 1313 =+=+ y xand y x .

10. The line through the points (h, 3) and (4, 1) i ntersects the line

7x 9 y 19 =0 at right angle. Find the va lue of h.

11. Prove that the line through the point (x 1 , y 1) and parallel t o the line

Ax + By + C = 0 is A (x x 1) + B (y y 1) = 0.

12. Two lines passing through the point (2, 3) intersects each other at an

angle o f 60 0 . If slope o f one l ine i s 2 , nd equation of the o ther line.

13. Find the equation of the right bisector of the line segment joining the

points (3, 4) and (1, 2).

14. Find the coordinates of the foot of perpendicular from the p oint (1, 3) to

the line 3 x 4 y 1 6 = 0.

15. The perpendicular from the origin to the line y = mx + c meets it at the

point (1, 2). Find the values of m and c.


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16. If p and q are the lengths of perpendiculars from the origin to the lines

x cos y sin = k cos 2 and x sec + y cosec = k, respectively, prove that p 2 + 4q 2 = k 2.

17. In the t riangle ABC with vertices A (2, 3), B (4, 1) and C (1, 2), nd the

equation and length of altitude f rom the vertex A.

18. If p is the length of perpendicular from the origin to the line whose

intercepts on the a xes a re a and b, then show that 222111

ba p+= .

M I C E L L A N E O U E X E R C I E

E.20.If the lines 2x + y 3 =0, 5x + ky 3 =0 and 3x y 2 = 0 areconcurrent, nd the value o f k.

E.21.Find the d istance of the line 4 x y =0 from the p oint P (4,1) measuredalong the line m aking a n angle of 135 0 w ith the positive x-axis.

E.22.Assuming that straight lines work as the plane mirror for a point, ndthe image of the p oint (1,2) in the l ine x 3 y + 4 = 0.

E.23.Show that the a rea of the t riangle formed by the lines.( )

21

221

2211 20,

mmcc

is xand c xm yc xm y

=+=+= .

E.24.A line is su ch that its se gment between the lines :5x y + 4 = 0 and 3x + 4y 4 =0 is b isected at the point (1,5). Obtain

its e quation.

E.25.Show that the path of a moving point such that its d istances from two

lines 3x 2 y =5 and 3x + 2y =5 are e qual is a straight line.

1. Find the values of k for which the line :

(k3) x (4 k 2) y + k 2 7k + 6 = 0 is

(a) Parallel to the x-axis,

(b) Parallel to the y -axis,

(c) Passing through the origin.

2. Find the values of and p, if the equation x cos + y sin = p is the

normal form of the l ine 3 x + y + 2 = 0.

3. Find the equations of the lines, which cut-off intercepts on the axes

whose su m and product are 1 and 6, respectively.


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4. What are the points on the y-axis w hose distance from the line 143

=+ y x

is 4 units.

5. Find perpendicular distance f rom the origin of the line joining the points

(cos, sin ) and (cos , sin ).6. Find the equation of the line parallel to y-axis a nd drawn through the

point of intersection of the lines

7. Find the equation of a line drawn perpendicular to the line 164

=+ y x

through the p oint, where i t meets t he y -axis.

8. Find the a rea of the triangle formed by the lines y x = 0, x + y = 0 and

x k = 0.

9. Find the value of p so t hat the three lines 3x + y 2 = 0, px + 2 y 3 = 0and 2x y 3 = 0 may intersect at one point.

10. If three lines w hose equations a re y = m 1x + c 1 , y = m 2x + c 2 and y

= m 3x + c 3 a re con current, then show that:

m 1(c 2 c 3) + m 2 (c 3 c 1) + m 3 (c 1 c 2) = 0.

11. Find the equation of the lines through the point (3, 2) which make an

angle o f 45 0 w ith the line x 2 y = 3.

12. Find the e quation of the line p assing through the p oint of intersection of

the lines 4 x + 7y 3 = 0 and 2x 3 y + 1 = 0 that has equal intercepts

on the a xes.

13. Show that the equation of the line passing through the origin and

making an angle w ith the line +=+=

tan1tan

mm

x y

iscmx y .

14. In what ratio, the line joining (1, 1) and (5, 7) is divided by the line

x + y = 4?

15. Find the d istance of the line 4x + 7y + 5 = 0 from the point (1, 2) along

the line 2 x y = 0.

16. Find the direction in which a straight line must be drawn through the

point (1, 2) so that its point of

be a t a distance o f 3 units f rom this p oint.

17. The hypotenuse of a right angled triangle has its ends at the points

(1, 3) and ( 4 , 1). Find the equation of the legs (perpendicular si des) of

the triangle if they are parallel to the axes.


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18. Find the image of the point (3, 8) with respect to the line x +3y = 7

assuming the line to be a plane m irror.

19. If the lines y = 3x +1 and 2y = x + 3 are equally inclined to the line

y = mx + 4, nd the value of m.

20. If sum of the p erpendicular distances of a variable p oint P (x, y) from the

lines x + y 5 = 0 and 3x 2y +7 = 0 is always 10. Show that P must

move on a line.

21. Find equation of the line which is equidistant from parallel lines

9x + 6y 7 = 0 and 3x + 2y + 6 = 0.

22. A ray of light passing through the point (1, 2) reects on the x-axis at

point A and the reected ray passes through the point (5,

coordinates of A.23. Prove that the product of the lengths of the perpendiculars drawn from

the points ( ( 0,0, 2222 baand ba to the line 21sincos bisb y

a x =+

.

24. A person standing at the junction (crossing) of two straight paths

represented by the equations 2x 3y + 4 = 0 and 3x + 4 y 5 = 0 wants

to reach the path whose equation is 6 x 7y + 8 = 0 in the least time.

Find equation of the p ath that he sh ould follow.


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Type II (Ext ! " !#t$#e Q%est$&ns)

1. Find the equation of a line which passes through the point

(2, 3) and makes an angle of 30 with the positive direction

of x -axis.

2. Find the equation of the line where length of the

perpendicular segment from the origin to the line is 4 and

the inclination of the perpendicular segment with the

positive direction of x -axis is 30.

3. Prove that every straight line has an equation of the form

A x + B y + C = 0, where A, B and C are constants.4. Find the equatio n of the straight line passing through (1, 2)

and perpendicular to the line x + y + 7 = 0.

5. Find the distance between the lines 3 x + 4 y = 9 and

6 x + 8 y = 15.

6. Show that the locus of the mid-point of the distance

between the axes of the variable line x cos + y sin = p is

222

411

p y x=+ where p is a constant.

7. If the line joining two points A(2, 0) and B(3, 1) is rotated

about A in anticlock wise direction through an angle of 15.

Find the equation of the line in new position.

8. If the slope of a line passing through the point A(3, 2) is 43

,

then nd points on the line which are 5 units away fromthe point A.

9. Find the equation to the straight line passing through the

point of intersection of the lines 5 x 6 y 1 = 0 and

3 x + 2 y + 5 = 0 and perpendicular to the line

3 x 5 y + 11 = 0.


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10. If one diagonal of a square is along the line 8 x 15 y = 0

and one of its vertex is at (1, 2), then nd the equation of

sides of the square p assing through this vertex.

11. Find the equation of the straight line which passes throughthe point (1, 2) and cuts off equal intercepts from axes.

12. Find the equation of the line passing through the point (5,

2) and perpendicular to the line joining the points (2, 3)

and (3, 1).

13. Find the angle between the lines y = (2 3 ) ( x + 5) and

y = (2 + 3 ) ( x 7).

14. Find the equation of the lines which passes through the

point (3, 4) and cuts off intercepts from the coordinate axes

such that their sum is 14.

15. Find the points on the line x + y = 4 which lie at a unit

distance from the line 4 x + 3 y = 10.

16. Show that the tangent of an angle between the lines

1=+b y

a x and 1= b

ya x is 222 ba

ab .

17. Find the equation of lines passing through (1, 2) and

making angle 30 with y -axis.

18. Find the equation of the line passing through the point of

intersection of 2 x + y = 5 and x + 3 y + 8 = 0 and

parallel to the line 3 x + 4 y = 7.

19. For what values of a and b the intercepts cut off on the

coordinate axes by the line ax + by + 8 = 0 are equal in

length but opposite in signs to those cut off by the line 2 x

3 y + 6 = 0 on the axes.

20. If the intercept of a line between the coordinate axes is

divided by the point (5,4) in the ratio 1 : 2, then nd the

equation of the line.


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21. Find the equation of a straight line on which length of

perpendicular from the origin is four units and the line

makes an angle of 120 with the positive direction

of x -axis.22. Find the equation of one of the sides of an isosceles right

angled triangle whose hypotenuse is given by 3 x + 4 y = 4

and the opposite vertex of the hypotenuse is (2, 2).

23. If the equation of the base of an equilateral triangle is

x + y = 2 and the vertex is (2, 1), then nd the length of

the side of the triangle.

[Hint: Find length of perpendicular (p) from (2, 1) to the

line and use p = l sin 60, where l is the length of side of

the triangle].

24. A variable line passes through a xed point P. The algebraic

sum of the perpendiculars drawn from the points (2, 0), (0,

2) and (1, 1) on the line is zero. Find the coordinates of the

point P.25. In what direction should a line be drawn through the point

(1, 2) so that its point of intersection with the line x + y = 4

is a t a distance36 from the given point.

26. A straight line moves so that the sum of the reciprocals of

its intercepts made on axes is constant. Show that the line

passes through a xed point.27. Find the equation of the line which passes through the

point ( 4, 3) and the portion of the line intercepted

between the axes is divided internally in the ratio 5 : 3 by

this point.


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28. Find the equations of the lines through the point of

intersection of the lines x y + 1 = 0 and 2 x 3 y + 5 = 0

and whose distance from the point (3, 2) is 57

.

29. If the sum of the distances of a moving point in a plane

from the axes is 1, then nd the locus of the point.

30. P 1 , P 2 are points on either of the two lines y 3 x = 2 at a

distance of 5 units from their point of intersection. Find

the coordinates of the foot of perpendiculars d rawn from P 1 ,

P 2 on the bisector of the angle between the given lines.

31. If p is the length of perpendicular from the origin on the

line 1=+ b y

a x

and 222 ,, b pa are in A.P, then show that

044 =+ba .32. Find the equation of the line for which p = 5 and 0135= . Also

sketch the line.

33. A line forms a triangle in the rst quadrant with coordinate axes.

If the area of the triangle is354

sq. units a nd the perpendiculardrawn from the origin to the line makes an angle 60 0 w ith x-axis,

nd the equation of the line.

34. Sketch roughly the lines sa tisfying the given conditions and write

their eq uations :

(i) inclination = 150 0 and distance from origin = 3.(ii) x-intercept = 7 and distance from the origin=2.

(iii) nearest point to the origin is (3, 4).

35. Find the equation of a line which passes through the point(2, 3) and makes angle 60 0 w ith the positive direction of x-axis.

36. Find the equation of the straight line which passes through the

point (2,9) and making an angle of 45 0 with x-axis. Also nd the

points on the line which are at the distance of (i) 2 units

(ii) 5 units from (2,9).

37. The line joining two points A (2,0) and B (3,1) is rotated about A

in anticlockwise direction through an angle of 15 0 . Find the


19


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equation of the line in the n ew position. If B goes to C in the new

position, what will be the coordinate of C?

38. Find the distance of the point (2,5) from the line 3x + y + 4 =0

measured parallel to a line having slope 3/4.

39. Find the direction in which a straight line must be drawn through

the point (1,2) so that its point of intersection with the line

x + y =4 may be at a distance of 3/6 unit from this point.

40. Find the equation of the line passing through (3,5) and (1,2),

assuming the equation of the line to be Ax + By + C =0.

41. Reduce 023 =++ y x to the 'slope-intercept from' and hence nd

its slope, inclination and y-intercept. Also sketch the line on the

coordinate plane.42. Reduce 4x 3 y 12 =0 to the 'intercept form' and hence nd its

intercepts on the axes. Also sketch the line on the coordinate

plane.

43. Reduce 043 =+ y x to the 'normal form' and hence nd the

values of p and . Also sketch the line on the coordinate p lane.44. Find the angle between the lines 3x + y 7 = 0 and

x + 2 y + 9 =0.

45. Find the angle between the lines:

( ) ( ) ( ) ( ) 0,322322 >>+=+++= bawherea xbab yaaband b xbab yaba .46. Find the angle between the lines joining the points (0, 0), (2,3)

and (2, 2), (3,5).

47. Prove that the points (2, 1), (0,2), (3,3) and (5,0) are the vertices

of a parallelogram. Also nd the a ngle between the d iagonals.

48. The acute angle between two lines is /4 and slope of one ofthem is . Find the slope of the other line.

49. The slope of a line is double of the slope of another line. If

tangent of the angle between them is 1/3, nd the slopes of the

lines.

50. Find the equation of the straight line which passes through the

origin and making angle 60 0 with the line 0333 =++ y x .


20


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51. Find the equations of the lines which pass through (4,5) and

make equal angles with the lines 5x 12 y + 6 =0 and

3x = 4y +7.

52. Show that the equation of the line through the origin and makingan angle w ith the line y = mx + c are

y = xmm

yand xm

m

tan1tan

tan1tan

+=

+.

53. The base of an equilateral triangle is the line x + y 2 =0 and the

vertex is the point (2, 1). Find the equations of the remaining

sides.

54. Two opposite vertices of a square are (3,4) and (1, 1). Find the

coordinates o f other vertices.

55. On the portion of the line x + 3y 3 =0 which is intercepted

between the coordinate axes, a square is constructed on the side

of the line away from the origin. Find coordinates of the

intersection of its d iagonals. Also, nd the e quations o f its s ides.

56. Find the equation of the straight line that has y -intercept 4 and is

parallel to the straight line 2x 3y =7.

57. Prove that the line through the point (x 1 , y 1 ) and parallel to the

line Ax + by + C =0 is A (x x 1 ) + B (y y 1 ) =0.

58. Find the equation of the straight line passing through (2,3) and

perpendicular to 4x 3y =10.

59. The line 7x 9 y 19 =0 is perpendicular to the line through the

points (x,3) and (4,1). Find the value of x.

60. Assuming the straight lines work as the plane mirror at a point,

nd the image of the point (1,2) in the line x 3y +4 =0.

61. The point (1,2) is the foot of the perpendicular from the origin to a

straight. Find the equation of this straight line.

62. A line perpendicular to the line segment joining the points (1,0)

and (2,3) divides it in the ration 1 : n. Find the equation of the

line.


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63. One side of a rectangle lies along the line 4x + 7y + 5 =0. Two of

its vertices are (3,1) and (1,1). Find the equation of the other

three sides.

64. The points (1,3), (5,1) are the opposite vertices of a rectangle. Theother two vertices lie on the line y = 2x + c. Find c and the

remaining vertices.

65. Find which of the following pairs of lines are intersecting,

parallel or coincident:

(i) 2x y +7 =0 and 2x + y 9 =0

(ii) x + 6y +11 = 0 and 2x +12y = 22

(iii) 3x y + 6 =0 and 2y 6x + 11 = 0.66. Find the equation of the straight line which passes through the

intersection of the lines x + y 3 =0 and 2x y =0 and is inclined

at an angle 45 0 w ith x-axis.

67. A person standing at the junction (crossing) of two straight paths

represents by the equations 2x 3y 4 =0 and 3x + 4y 5 =0,

wants to reach the path whose equation is 6x 7 y + 8 =0 in the

least time. Find the equation of the path that he should follow.68. Two lines cut the axis of x at distances of 4 and 4 and the axis

of y at distances 2 and 6, respectively. Find the coordinates of

their p oint of intersection.

69. Find the distance of the line 4x y =0 from the point (4,1)

measured along the line making an angle of 135 0 with the

positive direction of the x-axis.

70. Find the coordinates of the foot of perpendicular drawn from the

point (2,3) on the line 3x + 4y + 8 =0.

71. Show that the area of the triangle formed by the lines whose

equations are :

( )||2

0,21

221

2211 mmcc

is xand c xm yc xm y

=+=+= sq. units.

72. For what value of k, are the three lines:

0950352,012 =+=+=+ k y xand y x y x are concurrent?


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73. If a,b,c are in A.P., prove that lines a x + 2y +1 = 0, bx + 3y +1 =0

and cx+4y+1=0 are concurrent.

74. Find the coordinates of the orthocentre of the triangle whose

vertices are (0,1), (1, 2) and (2, 3).75. Find the coordinates of the circumcentre of the triangle whose

vertices are (4,3), (2,3) and (6, 1).

76. Find the length of perpendicular from (4,2) to the line

5x 12 y =0.

77. Find the distance between the line 9x + 40 y 20 =0 and

9x + 40 y + 21 =0.

78. Which of the line 2x + 7 y 9 =0 and 4x y +1 1 =0 is farther fromthe point (2,3)?

79. If p and q be the lengths of perpendiculars from origin to the

lines x sec + y cosec =k and x cos y sin = k cos 2 respectively, then prove that 4p 2 + q 2 = k 2 .

80. Find the point on the line x + y +9 =0 whose distance from the

line x + 3y 8 =0 is 103 .

81. A straight line passes through the point (1,2) and its distance from the origin is one unit. Find its equation.

82. If sum of the perpendicular distances of a variable point P (x,y)

from the lines x + y 5 =0 and 3x 2y + 7 =0 is always 10. Show

that P must move on a line.

83. Find the equation of the straight line passing the through the

intersection of 3x + 4y =7 and x y +2 =0 and with slope 5.

84. Find the equation of line parallel to the y-axis a nd drawn through

the point of intersection of x 7y + 5 = 0 and 3x + y 7

=0.

85. Find the equation of the line passes through the point of

intersection of the lines 4x + 7y 3 =0 and 2x 3y +1 =0, that

has equal intercepts o n the axes.

86. Find the equation of the straight line drawn perpendicular to the

line b y

a x

+ =1 through the point, where it meets the y-axis.


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87. Find the equation of line passing through the intersection of the

line x 3y +1 =0 and 2x + 5y 9 =0 and whose distance from the

origin is 2.

88. If t 1 and t 2 a re the roots of the equation t2

+ t + 1 =0, where isan arbitrary constant, then prove that the line joining the points

( )121 2, at at and ( )222 2, at at always passes through a xed point. Also nd that point.

89. Find the new coordinates of the following points if the origin is

shifted to (3,2) under a translation :

(i) (1,1) (ii) (2,1) (iii) (5,0) (iv) (1,2)

90. Transform the equation 0301843 22

=++++ y x y x when the axes aretranslated so that the n ew origin is (2, 3).

91. What does the equation (a b) (x 2 + y 2) 2abx =0 become if the

origin is shifted to the point

0,

baab

without rotation?

92. On shifting the origin to the point (1, 1), the axes remaining

parallel to the original axes, the equation of curve becomes

02434 22

=+++ y x y x . Find the original equation.93. Find the point to which the origin should be shifted after a

translation of axes so that the equation x 2+y 25x + 2y 5 =0 has

no rst degree terms.

94. Verify that the area of triangle with vertices (4,1), (6, 7) and

(0,5) remain invariant under the translation of axes when the

origin is shifted to the point (2, 4).

95. Prove that the area of a triangle is invariant under thetranslation of a xes.

96. Determine x so that the line passing of their slopes is 1. If m is

the slope of a line, the positive direction of x-axis.

97. Find the angle between the lines joining the points (0,0), (2,3)

and the points (2,2), (3,5).

98. Let A (6,4) and B (2,12) be two given points. Find the slope of a

line perpendicular to AB.


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99. Determine x so that 2 is the slope of the line through (2,5) and

(x,3).

100. What is the value of y so that the line through (3,y) and (2,7) is

parallel to the line through (1,4) and (0,6)?101. Without using Pythagoras theorem, show that A (4,4), B(3,5) and

C (1,1) are the vertices of a right-angled triangle.

102. A quadrilateral has the vertices at the points (4,2); (2,6), (8,5)

and (3,5). Show that the mid-points of the sides of this

quadrilateral are the vertices o f a parallelogram.

103. Prove that A (4,3), B (6,4), C (5,6) and D (3,5) are the angular

points of a square.104. If the angle between two line in

4

and slope of one of the line21

, nd the s lope of the other line.

105. If the points P (h,k), Q (x 1 , y 1 ) and R (x 2 , y 2 ) lie on a line, show

that :

106. A ray of light passing through the point (1,2) reects on the

x-axis at point A and the reected ray passes through the point

(5,3). Find the co-ordinates o f A.

107. Prove that the line joining the mid-points of the two sides of a

triangle is parallel to the third side.

108. If A (2,0), B (0,2) and C (0,7) are three vertices, taken in order, of

an isosceles trapezium ABCD in which AB || DC. Find the

coordinates of D.

109. If points (a, 0), (0,b) and (x, y) are collinear, using the concept of

slope p rove that 1=+b y

a x

.

110. By using the concept of slope, prove that the diagonals of a

rhombus a re a t right angles.

111. Using the concept of slope, prove that medians of an equilateral

triangle a re perpendicular to the corresponding sides.

112. Prove that a triangle which has one of the angle as 30 0, cannot

have all vertices with integral coordinate.


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113. The vertices of a triangle a re A (x 1 , x 1 tan 1 ), B (x 2 , x 2 tan 2 ) andC (x 3 , x 3 tan 3 ). If the circumcentre of ABC coincide with theorigin and H ( ) y x , is the orthocenter, show that

321

321

coscoscossinsinsin ++ ++= x y .


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Type III (O' e#t$ e Q%est$&ns)

1. If two vertices of an equilateral triangle have integral coordinates then

the third vertex will have

(a) integral coordinates

(b) coordinates which are rational

(c) at least one coordinate irrational

(d) coordinates w hich are irrational

2. The polar coordinates of the vertices of a triangle are (0, 0), ( )2/,3 and ( )6/,3 . Then the t riangle is(a) right angled (b) isosceles

(c) equilateral (d) none of these

3. A point moves in the x-y plane such that the sum of its distances from

two mutually perpendicular lines is always equal to 3. The area

enclosed by the locu s of the point is

(a) 2unit18 (b) 2unit

2

9

(c) 2unit9 (d) None of these.

4. If vertices P, Q, R, R of a triangle PQR are rational points then which of

the following point of the triangle PQR may not be a rational point.

(a) Centriod (b) Circumcentre(c) Orthocentre (d) incentre

5. The coordinates of the four vertices of a quadrilateral are (2, 4), (1,

2), (1, 2) and(2, 4) taken in order. The equation of the line passing

through the vertex (1, 2) and dividing the quadrilateral in two equal

areas is

(a) x + 1 = 0 (b) x + y = 1

(c) x y + 3 = 0 (d) none of these

6. The foot of the perpendicular on the line

=+ y x3

drawn from theorigin is C. If the line cuts the x-axis and y-axis at A and B

respectively then BC : CA is

(a) 1 : 3 (b) 3 : 1 (c) 1 : 9 (d) 9 : 1

7. The distance of the line 432 = y x from the point (1, 1) in the directionof the line 1=+ y x is

(a) 2 (b) 5 2 (c)2

1(d) none


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8. The coordinates of two consecutive vertices A and B of a regular

hexagon ABCDEF are (1, 0) and (2, 0) respectively. The equation of the

diagonal CE is

(a) 3x + y = 4 (b) x + 3y + 4 = 0(c) x + 3y = 4 (d) none of these

9. ABC is an isosceles triangle in which A is ( )0,1 , 3/2 = A , AB = AC and AB is along the x-axis. If BC = 4 3 then the equation of the

line BC is

(a) x + 3y = 3 (b) 3x + y = 3(c) x + y = 3 (d) None of these.

10. The graph of the function ( ) ( )1cos2cos.cos 2 ++ x x x is a(a) straight line passing the point ( )1sin,0 2 with slope 2(b) straight line passing through the origin

(c) parabola with vertex ( )1sin,1 2(d) straight line passing through the point ( )1sin,2/ 2 and

parallel to the x-axis

11. The limiting position of the point of intersection of the lines 143 =+ y x

and ( ) 231 2 =++ yc xc a s c t ends to 1 is(a) (5, 4) (b) (5, 4)

(c) (4, 5) (d) none of these

12. If a vertex of an equilateral triangle is the origin and the side oppositeto it has the equation 1=+ y x then the orthocenter of the t riangle is

(a)

31

,31

(b)

32

,32

(c)

3

2,

3

2(d) None of these

13. Let the perpendiculars from any point on the line 5112 =+ y x upon thelines 20724 =+ y x and 234 = y x have the lengths p and p

respectively. Then(a) p p =2 (b) p p = (c) p p = 2 (d) None

14. If ( )2/2,2/1 t t P ++ be any point on a line then the range of values oft for w hich the point P lies between the parallel lines 12 =+ y x and

1542 =+ y x is

(a)6

253

24


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(b) always passes through a xed point

(c) always cuts intercepts on the axes such that their sum is zero

(d) forms a t riangle with the axes whose area is constant


29


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16. If a, b, c are in GP then the line ax + b y + c = 0

(a) has a xed direction

(b) always passes through a xed point

(c) forms a triangle with the axis w hose area is constant

(d) always cuts intercepts on the axes such that their sum is zero17. A family of lines is given by ( ) ( ) 0121 =+++ y x , being the

parameter. The line belonging to this family at the maximum distance

from the point (1, 4) is

(a) 014 =+ y x (b) 071233 =++ y x

(c) 73312 =+ y x (d) none of these18. If the point (a, a) falls between the lines 2=+ y x then

(a) 2=a (b) 1=a (c) 1


32/50


(a)

2

3,

2

(b)

2,

2

(c) ( ) ,0 (d) none


31


33/50


24. The points (1, 1) and (1, 1) are symmetrical about the line

(a) 0=+ x y (b) x y = (c) 1=+ y x (d) none25. The equation of the line segment AB is x y = . If A and B lie on the

same side of the line mirror 12 = y x , the image of AB has theequation

(a) 2=+ y x (b) 98 =+ y x (c) 67 = y x (d) none26. Let ( )1,1= P and Q = (3, 2). The point R on the x-axis such that

Q P + is the minimum is

(a)

0,35

(b)

0,31

(c) ( )0,3 (d)

none

27. If a ray travelling along the line x = 1 gets reected from the line x

+ y = 1 then the equation of the line along which the reected raytravels is

(a) y = 0 (b) x y = 1 (c) x = 0 (d) none

28. The point P(2, 1) is shifted by 23 parallel to the line 1=+ y x , in the

direction of increasing ordinate, to reach Q. The image of Q by the line1=+ y x is

(a) (5, 2) (b) ( )2,1 (c) (5, 4) (d) ( )4,1

29. Let ( )0,1= A and ( )1,2= B . The line AB turns about A through an

angle 6/ in the clockwise sense, and the new position of B is B .Then B has the coordinates(a)

+

213

,2

33(b)

+

213

,2

33

(c)

+

231

,2

31(d) none of these

30. Two points A and B move on the x-axis and the y-axis respectively

such that the distance between the two points is a lways the sa me. The

locus o f the middle point of AB is(a) a straight line (b) a pair of straight lines

(c) a circle (d) none of these.

31. A variable line through the point (a, b) cuts the axes of reference at Aand B respectively. The lines through A and B parallel to the y-axisand the x-axis respectively meet at P. Then the locus of P has theequation

(a) 1=+b y

a x

(b) 1=+a y

b x

(c) 1=+ yb

xa

(d) 1=+ ya

xb

32. The equation 054 22 =++ y xykx represents two lines inclined at an angle if k is


32


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(a)45 (b)

54 (c)

54 (d) none

33. The equation 033 =+ y x represents(a) three real straight lines(b) three points(c) the combined equation of a straight line and a circle(d) none of these.

34. If the slope of one line is double the slope of another line and the

combined equation of the pair of lines is 02 22 =++

b y

h xy

a x

then 2: hab

is e qual to(a) 9 : 8 (b) 3 : 2 (c) 8 : 3 (d) none

35. The triangle formed by the lines whose combined equation is

( ) ( ) 014 22

=+ y x x xy y is(a) equilateral (b) right angled(c) isosceles (d) obtuse angled

36. The three lines whose combined equation is 04 23 = y x y form atriangle w hich is(a) isosceles (b) equilateral(c) right angled (d) none of these

37. The equation ! y x y x =+++ 3518832 22 represents(a) no locus if 0> ! (b) an ellipse if 0< ! (c) a point if K = 0 (d) a hyperbola if 0> !

38. The area of the triangle formed by two rays whose combined equation is x y = and the line 22 =+ y x is

(a) 2unit3

8 (b) 2unit3

4 (c) 2unit4 (d) 2unit3

16

39. The centroid of the triangle whose three sides are given by thecombined equation ( ) ( ) 0127 22 =++ y y xy x is(a)

0,3

2 (b)

3

2,

3

7 (c)

3

2,

3

7 (d) None

40. The orthocenter of the triangle formed by the pair of lines0122 22 =++ y x y xy x and the line 01 =++ y x is

(a) ( )0,1 (b) (0, 1) (c) (1, 1) (d) none41. The angle between the pair of lines whose equation is

0105104 22 =++++ y xmy xy x is

(a)8

3tan 1 (b)

4

3tan 1

(c) m

m

m

+

,4

4252tan 1 (d) none of these


33


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42. The combined equation of the pair of lines through the point (1, 0) and parallel to the lines represented by 02 22 = y xy x is

(a) 6422 22 =+ y x y xy x

(b) 0242 22 =+ y x y xy x

(c) 0242 22 =++ y x y xy x(d) none of these

43. The pair of lines 3 2 x 4xy + 3 2 y = 0 are rotated about the origin by

6

in the anticlockwise sense. The equation of the pair in the new

position is

(a) 3 2 x xy = 0 (b) 2 x 3xy = 0(c) xy 3 2 y = 0 (d) none of these

44. The equation of the image of the pair of rays x y = by the line 1= x is(a) 2+= x y (b) x y =+ 2(c) 2= x y (d) none of these

45. Two lines represented by the equation 01222 =+ x y x are rotatedabout the point (1, 0) the line making the bigger a ngle w ith the positivedirection of the x-axis b eing turned by 45 in the clockwise sense andthe other line being turned by 15 in the anticlockwise sense. Thecombined equation of the pair of lines in their new positions is(a) 03323 2 =++ y x xy x

(b) 03323 2

=++ y x xy x

(c) 03323 2 =+ x xy x(d) none of these

46. x y 10= is the re ection of x y 10log= in the line w hose equation is(a) y = x (b) y = x (c) y = 10x (d) y = 10x

47. Let PQR be a right angled isosceles triangle, right angled at P(2, 1). Ifthe equation of the line QR is 32 =+ y x , then the equation representingthe lines PQ and PR is(a) 0251020833 22 =++++ y x xy y x(b) 0251020833 22 =++ y x xy y x

(c) 0201510833 22

=++++ y x xy y x(d) 0201510833 22 = y x xy y x

48. Line L has intercepts a and b on the coordinate axes. When the axesare rotated through a given angle, keeping the origin xed, the sameline L has intercepts p and q, then

(a) 2222 q pba +=+ (b) 22221111

q pba+=+

(c) 2222 qb pa +=+ (d) 22221111

qb pa+=+

49. The orthocentre of triangle formed by the lines xy = 0 and x + y = 1 is

(a)

21,

21 (b)

31,

31 (c) ( )0,0 (d) 4

1,41


34


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50. The diagonals of parallelogram PQRS are along the lines 43 =+ y x and726 = y x . Then PQRS must be

(a) rectangle (b) square

(c) cyclic quadrilateral (d) rhombus


35


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51. The inclination of the line x y + 3 = 0 with the positive directionof x-axis is(a) 45 (b) 135 (c) 45 (d) 135

52. The two lines ax + b y = c and a'x + b'y = c'are perpendicular if

(a) aa'+ bb'= 0 (b) ab'= ba'(c) ab + a b'= 0 (d) ab'+ ba'= 0

53. The equation of the line passing through (1, 2) and perpendicularto x + y + 7 = 0 is(a) y x + 1 = 0 (b) y x 1 = 0(c) y x + 2 = 0 (d) y x 2 = 0.

54. The distance of the point P (1, 3) from the line 2y 3x = 4 is

(a) 13 (b) 13137

(c) 13 (d) none of these

55. The coordinates of the foot of the perpendicular from the point (2,3) on the line x + y 11 = 0 are(a) (6, 5) (b) (5, 6) (c) (5, 6)(d) (6, 5)

56. The intercept cut off by a line from y-axis is twice than that fromx-axis, and the line passes through the point (1, 2). The equationof the line is(a) 2x + y = 4 (b) 2x + y + 4 = 0(c) 2x y = 4 (d) 2x y + 4 = 0

57. A line passes through P (1, 2) such that its intercept between theaxes is bisected at P. The equation of the line is(a) x + 2y = 5 (b) x y + 1 = 0(c) x + y 3 = 0 (d) 2x + y 4 = 0

58. The reection of the point (4, 13) about the line 5x + y + 6 = 0is

(a) ( 1, 14) (b) (3, 4)(c) (0, 0) (d) (1, 2)

59. A point moves such that its d istance from the point (4, 0) is halfthat of its distance from the line x = 16. The locus of the point is(a) 3x2 + 4y2 = 192 (b) 4x2 + 3y2 = 192(c) x2 + y2 = 192 (d) None of these

60. A line cutting off intercept 3 from the y-axis and the tengent at

angle to the xaxis is5

3 , its equation is

(a) 5y 3x + 15 = 0 (b) 3y 5x + 15 = 0(c) 5y 3x 15 = 0 (d) None of these

61. Slope of a line which cuts off intercepts of equal lengths on theaxes is

(a) 1 (b) 0 (c) 2 (d) 3


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62. The equation of the straight line passing through the point (3, 2)and perpendicular to the line y = x is(a) x y = 5 (b) x + y = 5(c) x + y = 1 (d) x y = 1


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63. The equation of the line passing through the point (1, 2) and perpendicular to the line x + y + 1 = 0 is

(a) y x + 1 = 0 (b) y x 1 = 0(c) y x + 2 = 0 (d) y x 2 = 0

64. The tangent of angle between the lines whose intercepts on theaxes are a, b and b, a , respectively, is

(a)ab

ba 22 (b)

2

22 ab

(c)ab

ab2

22 (d) none of these

65. If the line 1=+b y

a x

passes through the points (2, 3) and (4, 5),

then (a, b) is :(a) (1, 1) (b) (1, 1)

(c) (1, 1) (d) (1, 1)66. The distance of the point of intersection of the lines 2x 3 y + 5 =

0 and 3x + 4y = 0 from the line 5x 2y = 0 is:

(a)2917

130(b)

297

13

(c)7

130(d) none of these

67. The equation of the lines which pass through the point (3, 2)and are inclined at 60 0 to the line 3 x + y = 1 is :

(a) 03323,02 ==+ y x y(b) 03323,02 =++= y x x

(c) 03323 = y x

(d) none of these68. The equation of the lines passing through the point (1, 0) and at

a distance2

3 from the origin, are :

(a) 033,033 ==+ y x y x

(b) 033,033 =+=++ y x y x

(c) 033,033 ==+ y x y x

(d) none of these69. The distance between the lines y = m x + c 1 and y = mx + c 2 is :

(a)12

21

+

m

cc(b) 2

21

1

||

m

cc

+

(c) 212

1 m

cc

+

(d) 0

70. The coordinates of the foot of perpendiculars from the point (2,3)on the line y = 3x + 4 is g iven by:

(a)

10

1,

10

37

(b)

10

37,

10

1

(c)

10,37

10

(d)

3

1,

3

2


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71. If the coordinates of the middle point of the portion of a line

intercepted between the coordinate axes is (3,2), then the

equation of the line will be :

(a) 2x + 3y = 12 (b) 3x + 2y = 12(c) 4x 3y =6 (d) 5x 2y = 10

72. Equation of the line passing through (1,2) and parallel to the line

y = 3x 1 is :

(a) y + 2 = x + 1 (b) y + 2 = 3 (x + 1)

(c) y 2 = 3 (x 1) (d) y 2 = x 1

73. Equations of diagonals of the square formed by the lines

x = 0, y = 0, x = 1 and y = 1 are :(a) y = x, y + x = 1 (b) y = x, x + y = 2

(c) 2y = x, y + x =3

1(d) y = 2x, y + 2x = 1

74. For specifying a straight line, how many geometrical parameters

should be known?

(a) 1 (b) 2 (c) 4 (d) 3

75. The point (4,1) undergoes the following two successive

transformations :

(i) Reection about the line y = x

(ii) Translation through a distance 2 units along the positive x-

axis

Then the nal coordinates of the point are:

(a) (4,3) (b) (3,4) (c) (1,4) (d)

2

7,

2

7

76. A point equidistant from the lines4x + 3y + 10 = 0, 5x 12y + 26 = 0 and 7x + 24y 50 = 0 .

(a) (1, 1) (b) (1,1) (c) (0,0) (d) (0,1)

77. A line passes through (2,2) and is perpendicular to the line

3x + y = 3. Its yintercept is :

(a)3

1(b)

3

2(c) 1 (d)

34

78. The ratio in which the line 3x + 4y + 2 = 0 divides the distance

between the lines 3x + 4 y + 5 = 0 and 3x + 4 y 5 = 0 is :


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(a) 1 : 2 (b) 3 : 7 (c) 2 : 3 (d) 2 : 5


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80. Fill in the blanks :

(a) If a, b, c a re in A.P., then the straight lines ax + by + c = 0 will always

pass through _____________.

(b) The line which cuts off equal intercept from the axes a nd pass through

the point (1, 2) is ________________.

(c) Equations of the lines through the point (3, 2) and making an angle of

45 with the l ine x 2 y = 3 are _______________.

(d) The points (3, 4) and (2, 6) are s ituated on the _____________________

of the line 3 x 4 y 8 = 0.

(e) A point moves so that square of its distance from the point (3, 2) is

numerically equal to its distance from the line 5x 12y = 3. The

equation of its locus is ____.(f) Locus of the mid-points of the portion of the line

x sin + y cos = p intercepted between the axes is

____________.

81. State whether the statements True or False

(a) If the vertices of a triangle have integral coordinates, then the

triangle ca n not be e quilateral.

(b) The points A ( 2, 1), B (0, 5), C ( 1, 2) are collinear.

(c) Equation of the line passing through the point (a cos3, a sin3)

and perpendicular to the line x sec + y cosec = a is x cos

y sin = a sin 2.

(d) The straight line 5x + 4y = 0 passes through the point of

intersection of the straight lines x + 2y 10 = 0 and

2x + y + 5 = 0.

(e) The vertex of an equilateral triangle is (2, 3) and the equation of

the opposite side is x + y = 2. Then the other two sides arey 3 = (2 3 ) (x 2).

(f) The equation of the line joining the point (3, 5) to the point of

intersection of the lines 4x + y 1 = 0 and 7x 3y 35 = 0 is

equidistant from the points (0, 0) and (8, 34).

(g) The line 1=+b y

a x

moves in such a way that 222111

cba=+ , where c i s

a constant. The locus of the foot of the perpendicular from the

origin on the given line is x2

+ y2

= c2

.


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(h) The lines ax + 2y + 1 = 0, bx + 3y + 1 = 0 a nd cx + 4y + 1 = 0 are

concurrent if a, b, c a re in G.P.

(i) Line joining the points (3, 4) and ( 2, 6) is p erpendicular to the

line joining the points (3, 6) and (9, 18).


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82. Match the Columns:I.

Column C 1 Column C 2

(a) The coordinates of the

points P and Q on the linex + 5y = 13 which are at adistance of 2 units from theline 12x 5y + 26 = 0 are

(i) (3, 1), (7, 11)

(b) The coordinates of the point on the line x +y =4,l

which are at a unitdistance from the line4x + 3y 10 =0

(ii)

3

7,

3

4,

3

11,

3

1

(c) The coordinates of the point on the line joining A

(2, 5) and B (3,1) suchthat AP = PQ = QB are

(iii) 516,3,

512,1

II. The value of the , if the lines (2x + 3y + 4) + (6x y +12) = 0are :


(a) parallel to y-axis is (i) =4

3

(b) perpendicular to 7x + y 4=0 is

(ii) = -3

1

(c) passes through (1, 2) is (iii) = -41

17

(d) parallel to x axis is (iv) = 3III. The equation of the line through the intersection of the lines

2x 3y = 0 a nd 4 x 5y = 2 a nd :


(a) through the point (2,1) is (i) 2 x y = 4(b) perpendicular to the line x

+ 2y +1 = 0 is(ii) x + y 5 = 0

(c) parallel to the line 3x 4y+ 5 = 0

(iii)

x y 1 = 0

(d) equally inclined to the axesis

(iv) 3x 4y 1 = 0


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46/50


E x e r c i s e 1 0 . 2

E x e r c i s e 1 0 . 3


45


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M i s c e l l a e ! " s E x e r c i s e ! C % a $ & e r ' 1 0


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Type II (Ans,e s Ext ! p !#t$#e Q%est$&ns)

1. ( ) 02333 =+ y x 2. 83 =+ y x

4. x y 1 = 0 5. 103

7. 0323 =+ x y 8. (1, 1) or (7, 5)9. 5x + 3y +8 = 0 10. 7x + 23y 53 = 0

11. x + y + 1 = 0 12. x 4y + 3 = 0

13. 60 0 or 120 0 14. x + y = 7 or 186

=+ y x

15. (3, 1), (7, 11) 17. 0323 =+ x y

18. 3x + 4y + 3 = 0 19. 4,38 == ba

20. 8x 5y + 60 = 0 21. 83 =+ y x

22. x 7y 12 = 0 23.32

24. (1, 1) 25. 15 0 or 75 0

27. 9x 20y + 96 = 0 28. 3x 4y + 6 = 0 and 4x 3y +1 = 0

30.

+

235

2,0 32. 025 =+ y x

33. 0183 =+ y x34. (i) 063 =+ y x (ii) 014532 = y x

(iii) 3x 4y 25 = 0

35.2/3

32/12 =+ y x =r, where 'r' is t he d istance between (x,y) and (2,3)

36. (i) ( )29,22 ++ (ii)

++

2

59,

2

52

37. 0323 = y x ,

+26

,22

2 38. 5 units

39. 15 0 or 75 0 40. 7x 2y 11 = 0

41. 0120,2,3 52. 4

43. p =2 and = 30 0 44. = 45 0

45. 4422

1 4tanba

ba=

46. ( )23/11tan 1=

48. 1/3 or 3. 49. 1, 2,1,21

50. 0,03 == x y x 51. 4x 7y + 19 =0, 7x + 4y 48 =0.

53. ( ) ( ) 032532032532 =+=+ y xand y x


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54. D are (9/2, 1/2) and B are (1/2, 5/2)

55. Equation AD = 3x y 9 = 0, Equation of BC = 3x y +1 =0

56. 2x 3y + 12 = 0 58. 3x + 4y 18 = 0

59. x =9

2260. h =

56

, k =57

61. x + 2y 5 = 0 62. 0111

3 =+

++

nn

y x

63. AD = 7x 4y + 25 = 0, CD = 4x + 7y 11 = 0

BC = 7x 4y 3 = 0

64. (2, 0), (4, 4)

65. (i)2

1

2

1

B B

A A , so that lines are intersecting.

(ii)2

1

2

1

2

1

C

C

B

B

A

A == , so the lines are coincident.

(iii) ,2

1

2

2

2

1

C

C

B

B

A

A = s o the lines are parallel.

66. x y +1 = 0 67. 119x + 102y 205 = 0

68. (2,3) 69. 23 units.

70. (28/25, 29/25) 72. k = 4

74. (7, 6) 75. (1, 1) 76. 1

77. 1 78. 2.198, 3.881

80. (65/2, 47/2) and (5/2, 13/2)

81. 3x + 4y 5 = 0 and x = 1

83. 35x 7y + 18 = 0 84. x 2 = 0

85. 5x y = 0 and 13x + 13y 6 = 0

86. ax by + b 2 = 0 87. 3x + 4y 10 = 0, x 2 = 0

88. (a, 0)

89. (i) (4,3)(ii) 1,3) (iii) (8,2)(iv) (2,0)

90. X 2 + 3y 2 =1 91. ( ) ( ) 22222 baY X ba =+

92. 4X2

+ Y2

5X 2Y = 0


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93. (5/2, 1) 94. 12sq. units.

96. x = 2 97.

=

2311

tan 1

98. AB is 211

= m 99. x =1

100.y =9 101.1

104.3 or3

1 106. h =

513

108.x = 7 and y =0

49

straigt lines (2)

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