STRAIGHT LINES & CIRCLES

ASSIGNMENT -4 (DATE:- 16.04.2020) SINGLE CORRECT

1. Let p be a point in the plane, let 1d p and 2d p be the distances of the point p from lines 3 4 0x y

and 4 3 0x y respectively. Area of region R consisting of all points p lying in the first quadrant of the

plane and satisfying 1 21 3d p d p is

A) 2 B) 4 C) 6 D) 8

2. In a ABC , the measure of angle A is 075 and the measure of angle B is

045 and if D is a point on BC

such that AD = 4. If AD is the altitude then area of ABC is 1 and if AD is angular bisector then area of

ABC is 2 then 1 2 equals

A) 30 6 3 B) 8 6 3

C) 2 6 3 D) 20 6 3

3 In given figure find radius of circumcircle of PAB if 1 6r and 2 27r =

1)9

2 2)

92

2 3)9 2 4)None

4 Let P be a point lies inside the triangle ABC and D,E,F are feet of perpendiculars

from P to the lines BC,CA,AB respectively. If BC CA AB

PD PE PF is minimum then P is

A) Orthocentre of le ABCB) Circum centre of

le ABC

C) Incentre of le ABC D) Centroid of

le ABC

5 49. A circle with centre Q having double contact with parabola

( )2y 2 3 x= + passes through its focus S. If tangent at their point of contact P

intersect its axis at R, then radius of inscribed circle of triangle PSR is

A) 2 3 B) 3

4 C) 3 D)

3

2

MULTY CORRECT

6. If ,x y R then the equation 4 2 2 4 23 2 19 8 361 2 100 64 2 190 2x y x y y y y

represents in rectangular Cartesian system. Then which of the following can be valid locii?

A) parabola B) hyperbola C) circle D) ellipse

7. The triangle ABC has A 60 , B 45 . The bisector of A meets the side BC at T where AT=24. Then

which of the following is correct?

A) length of BC 18 2

B) length of AC AT

C) radius of the circum circle is 12 2

D) area of the triangle is 72 3 3 sq.units

8. Let A be a point on x-axis, B be a point on 2x-y+6=0 and C(2,5) such that the perimeter of ABCD is

minimum then which is/are correct

A) ( )1,4B = - B) A=(1,0)

C) Perimeter 2 10= D) Area of ABCD equals 20

3square units

9. If the circle 1C touches x-axis and the line tan , 0,2

y x

in first quadrant and circle 2C touches

the line tan ,y x y axis and circle 1C in such a way that ratio of radius of 1C to radius of 2C is 2 : 1,

then value of tan2

a b

c

where a,b,c are relatively prime natural numbers then which of the following

is/are true?

A) , ,a b c are prime numbers B) , ,a b c are not all prime numbers

C) 20a b D) 1b c

10. Circle 1 2 3S ,S ,S with radii 1 2 3r r r are such that i jS S 0 represents same line i j . 1S lies on one

side of that line & 2 3S ,S on another side. Line passing through centers 1 2 3O O O of circles 1 2 3S ,S ,S

intersects the circles 1 2 3S ,S ,S at 1 1 2 2 3 3A ,B ;A ,B ; A ,B respectively and the line i jS S at C, then which

of the following is/are CORRECT?

A) 1 1 2 2 3 3max min(CA , CB ), min CA , CB ,min CA , CB is 1CA or

1CB

B) 1 1 2 2 3 3max min(CA , CB ), min CA , CB ,min CA , CB is 3CA or

3CB

C) 1 1 2 2 3 3max max(CA , CB ), max CA , CB ,max CA , CB is 3CA or

3CB

D) 1 1 2 2 3 3max max(CA , CB ), max CA , CB ,max CA , CB is 1CA or 1CB

INTEGERS

11. If ABCD be a parallelogram whose diagonals have equations : 2 3AC x y+ = and

: 2 3BD x y+ = . If the length of diagonal AC=4 and area of ABCD equals 8 sq.units.

and The length of side AB can be

The length of side BC can be

p

q . Then

p

q

= ….. (where [.] = G.I.F)

12. If the system of equations ( 2ab h )

0ax hy g ---(1)

0hx by f ---(2)

and 2 22 2 2 0ax hxy by gx fy c t ---(3)

has unique solution, and 2 2 2

2

28

abc fgh af bg ch

h ab

, then t equals ...........

13.

14.

15.

PASSAGE TYPE

Paragraph for Question Nos. 16 to 17

Consider a ABC whose sides ,AB BC and CA are represented by the straight lines

2 0,x y x py q and 3x y respectively. The point P is 2,3

16. If P the centroid, then p q equals

A) 47 B) 50 C) 65 D) 74

17. If P is the orthocenter, then p q equals

A) 47 B) 55 C) 50 D) 65

PARAGRAPH FOR_18, 19

18.

19.

MATRIX MATCH

20. A B C, be an isosceles triangle with AB=AC.

If “AB” lies along the line x+y=10 and “AC” lies along the line 7x-y=30 and has area

20 square units then.

Column-I Column-II

I. Coordinates of “B” can be p. (4,-2)

II. Coordinates of “C” can be q. 19,1

3

III. Centroid of triangle ABC can be r. 25 5,

4 4

IV. Circum centre of triangle ABC, can be s. (10,0)

(A) I p II s III q IV r (B) I s II p III q IV r

(C) I s II p III r IV q

(D) I p II r III p IV s

21. Match the column:

Column-I Column-II

A) Line L is the radical axis of the circles P) 13

2 21 2 2 7 0S x y x y and 2 2

2 6 8 0S x y x y . If

1 1,x y and 2 2,x y denote the coordinates of the

extremeties of the diameter of 2S which is perpendicular

to L, then 2 2 2 21 2 1 2

1

5x x y y is equal to

B) The pair of lines represented by 2 2 3 4 1 0x y xy x y

intersect at P. If Q and R are the point of intersection of

the pair of lines with the x-axis and the area of the PQR

is , then 2

Q) 20

C) If the coordinates of radical centre of circles 2 2 25 0x y

; 2 2 2 22 2 4 6 7 0; 2 9 0x y x y x y x y is ,

then, 2 is equal to

R) 25

D) Let 1m and 2m are the slopes of the tangents drawn to

circle 2 2 4 8 5 0x y x y from the point ( 1, 2)P , and

1 2

pm m

q where p and q are relatively prime natural

numbers, then p q is equal to

S) 27

T) 29

KEY & SOL

1 2 3 4 5 6 7 8 9 10

B A 3 C D

ABC CD

AD ACD BD

11 12 13 14 15 16 17 18 19 20

5 8 8 4 3 D C A D B

21

A-

Q,B-

Q,C-

T,D-P

1. KEY:B

Sol: , 0x y

3 4 4 31 3

5 5

x y x y

5 3 4 4 3 15x y x y

Case-1: 3 4 0x y

5 3 4 4 15 15 7 15x y x x y

Case-2: 3 4 0x y

5 4 3 4 3 15 5 7 15y x x y x y

A ABCD ABPD

= 16 12

47 7

B

A

5,0

7P

15,0

7Q

12 9,

2 5B

150,

7C

50,

4C

3 4 0x y

2. KEY: A

SOL: 1 2 1 28 3 3 , 2 3 3 30 6 3

3 Key: 3

.

2

1 1sin 90 sin 2

r PB

1

2 2cos sin

r PAsimilarly

2

12sin

PBr

1

22sin

PAr

2

2 1

1 2

.. ( )

4sin .sin

PA PBr r R Ris required Radius

2 1R r r

4 KEY: C

Area of le ABC 1 1 1

... 12 2 2

BC x AC y AB z

Let BC CA AB

PD PE PF

a b c

x y z

Now 2a b c

ax by czx y z

A

B CD

x

py

z

2a b c

ax by czx y z

2 2 2 x y y z x za b c ab bc ac

y x z x z x

5 Key: d

N is centre of circle radiusSN SP PN

60PNS in PSR , 30 ,R PS SR

1

tan 30 3tt

3 ,2 3 , ,0 , 3 ,0P a a S a R a

Now,

1 3. . .

32 21 2

2

RS PS

rS

RS PS RP

MULTY CORRECT

6. Key :ABC

Sol: The given equation becomes 2 224 2 2 23 2 19 8 19 10 10 8 0x y x y y y

2 224 2 2 2 2 23 2 19 10 10 8 19 10 10 8 0x y y y x y y y

2 22 22 2 2 219 10 10 8 0x y x y x y

2 2 2 2 219 10, 10, 8x y x y x y

7 KEY: CD

8. KEY: AD

SOL: ( )' 2, 5C = -

( )'' 2,7C = -

2x-y

+6=

0

BC(2,5)

A

C(2,-5)

C(-2,7)

Perimeter AB+BC+CA

=AB+BC’’+AC’

' "C C³

16 144 4 10³ + =

' ''C C equation : 3x+y-1=0

Solving with ' ''C C

1

,03

Aæ ö

÷ç® = ÷ç ÷çè ø : B=(-1,4)

Area 3 3 / 51 1 5 20

151 52 2 3 3

æ ö÷ç= = - =÷ç ÷çè ø

sq. units

9. KEY: ACD

SOL:

r

2r

A

O/ 2

4 2

2 cot ,2

rOA

Let tan2

t

1 2 3 17

1 2

tt

t t

17 3

tan2 2

10. KEY: BD

SOL: Largest circle has its circumference, nearest to the radical axis

INTEGERS

11. key: 5

Sol:

12

32tan1 2 4

q- +

= =+

( )( )1 20

sin 22 3

PC PB BDq = Þ =

APBD , 2

583

AB =

,DPBD2

103

BC =

12. key: 8

Key:

13.8 14.4 15.3

13.

14.

15.

PASSAGE TYPE

16. Key: D

17. Key: C

Sol: 16, 17 . Consider a ABC whose sides ,AB BC and CA are represented by the straight lines

2 0,x y x py q and 3x y respectively. The point P is 2,3

(i) 1, 2 , , 2 , , 3A B C

1 6, 2 2 3 9 3, 8

3,6 , 8,5B C

Equation of BC is 11 63x y

63 11 74p q

(ii) Slope of 1BP 3 2

1 52

Slope of 1 6 1

102 2 2

CP

5,10 , 10,7B C

Equation of : 5 45BC x y

50p q

Key :- 18. A 19. D

18.

19.

MATRIX MATCH

20. Key: B

21. Key:-A-Q,B-Q,C-T,D-P

Sol:-Conceptual