circul and conic section (2)

8/11/2019 Circul and Conic Section (2)

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MATHEMATICSBY

Attri D.****** Mathematics by Attri. D. ******Mob : 9315820788******

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CIRCLE AND CONIC SECTION

****** Mathematics by Attri. D. ******Mob : 9315820788******

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nzks.kkpk;ZAcademy ResonantClasses

S.C. F. 57, Sector 7 S.C.O. 53, Sector 17Kurukshetra Kurukshetra

CIRCLE AND CONIC SECTION

11.1 Overview

11.1.1 Sections of a cone Let l be a fixed vert ical line and m be another line

intersecting it at a fixed point V and inclined to it at an angle (fig.).

Suppose we rotate the line m around the line l in such a way that the angle

remains constant. Then the surface generated is a double-napped right circular

hollow cone herein after referred as cone and extending indefinitely in both

directions (ig. !!.").

The point V is called the vertex# the line l is the l axis of the cone. The

rotating line m is called a generator of the cone. The generator vertex separates

the cone into two parts called nappes.


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$f we ta%e the intersection of a plane with a cone& the sect ion so obtained is

called a conic section. Thus& conic sections are the curves obtained by intersecting

a right circular cone by a plane.

'e obtain different %inds of conic sections depending on the position of theintersecting plane with respect to the cone and the angle made by it with the

vertical axis of the cone. Let be the angle made by the intersecting plane with the

vertical axis of the cone (ig.!!.).

The intersection of the plane with the cone can ta%e place either at the

vertex of the cone or at any other part of the nappe either below or above the

vertex. 'hen the plane cuts the nappe (other than the vertex) of the cone& we have

the following situations

(a) 'hen * +, o & the section is a circle(b) 'hen +, ,& the sect ion is an el lipse.

(c) 'hen * # the section is a parabola. ($n each of the above three

si tuations& the plane cuts entirely across one nappe of the cone).

(d) 'hen , # the plane cuts through both the nappes and the

curves of intersection is a hyperbola.

$ndeed these curves are important tools for present day exploration of outer

space and also for research into the behaviour of atomic part icles. 'e ta%e conic

sect ions as plane curves. or th is purpose& it is convenient to use euivalentdefinition that refer only to the plane in which the curve lies& and refer to special

points and lines in this plane called foci and directrices. /ccording to th is

approach& parabola& ellipse and hyperbola are defined in terms of a fixed point

(called focus) and fixed line (called directrix) in the plane.

$f S is the focus and l is the directrix& then the set of al l points in the plane

whose l distance from S bears a constant ratio e called eccentricity to their

distance from l is a l conic section.

/s special case of el lipse& we obtain circle for which e * , and hence we

study it di fferently.

!!.!." 0ircle / circle is the set of all points in a plane which are at a fixed

distance from a fixed point in the plane. The fixed point is called the centre of the

circle and the distance from centre to any point on the circle is called the radius of

the circle.

The euation of a circle with radius r having centre (h& %) is given by (x 1 h) " 2

(y 1 %)"* r"The general euation of the circle is given by x "2 y""gx 2 fy 2 c *

,& where g& f and c are constants.

(a) The centre of th is circle is (1g& 1 f)


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(b ) The radius of the circle is cfg + 22 .

The general euation of the circle passing through the origin is given by

x"2 y"2 "gx 2 "fy * ,.

3eneral euation of second degree i.e.&

ax "2 "hxy 2 by"2 "gx 2 "fy 2 c * , represent

a circle if

(i) the coefficient of x" euals the coefficient of

y"& i.e. & a * b 4, and

(ii) the coefficient of xy is 5ero& i .e.& h * ,.

The parametric euations of the circle x"

2 y"

* r"

are given by x * r cos& y * r sin where 4is the

parameter and the parametr ic euations of the

circle (x 1 h)"2 (y 1 %)"* r" are given by

x 1 h * r cos& y 1 % * r sin

6r x * h 2 r cos& y * % 2 r sin

7ote The general euation of the circle involves three constants which implies

that at least three conditions are reuired to determine a circle uniuely.

11.1.3 Parabola

/ parabola is the se t of points 8 whose distances from a

fixed point in the plane are eual to their distances from

a fixed line ! in the plane. The fixed point is called focus

and the fixed line ! the directrix of the parabola.

Standard equations of parabola

The four possible forms of parabola are shown below in ig. !!.9 (a) to (d)

The latus rectum of a parabola is a line segment perpendicular to the axis of the

parabola& through the focus and whose end points lie on the parabola (ig. !!.9).


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Focal distance of a point

Let the euation of the parabola be y "* :ax and 8(x& y) be a point on it. Then the

distance of 8 from the focus (a& ,) is called the focal distance of the point& i.e.&

11.1.4 Ellipse/n el lipse is the se t of points in a plane& the sum of whose distances from

two fixd points is constant. /lternatively& an ellipse is the set of all points in the

plane whose distances from a fixed point in the plane bears a constant ratio& less

than& to their distance from a fixed line in the plane. The fixed point is called

focus& the fixed line a directrix and the constant ratio (e) the centrici ty of the

ellipse. 'e have two standard forms of the ellipse& i.e.&

$n (i ) ma;or axis is along x-axis and minor along y-axis and in (i i) ma;or axis isalong y-axis and minor along x-axis as shown in ig. !!.< (a) and (b) respectively.


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orms of the el lipse1

2

2

2

2

=+

b

y

a

x1

2

2

2

2

=+

a

y

b

x

a = b a = b

>uation of manor axis y * , x * ,

Length of ma;or axis "a "a

>uation of ?inor axis x * , y * ,

Length of ?inor axis "b "b

directrices x * e

a

y *e

a

>uation of latus

rectum

x * ae y * ae

Length of latus rectum

a

b2

2

a

b22

0entre (,& ,) (,& ,)


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Type 1 !"E#T $%EST&O!S'

E(E#"&SE 11.1

>.!. ind an euation of the circle with centre at (,& ,) and radius r.>.". ind the euation of the circle with centre (1&") and radius :.

>.. ind the centre and the radius of the circle x "2y"2.@. ind the coordinates of the focus& axis& the euation of the directrix and latus rectum

of the parabola y"* .A. ind the euation of the parabola with focus ("&,) and directrix x * 1"

>.9. ind the euation of the parabola with vertex at (,&,) and focus at (,&")

>.


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&n eac) of t)e followin* E+ercise 1 to - find t)e coordinates of t)e focus- a+is of t)e

parabola- t)e equation of t)e directi+ and t)e len*t) of t)e latus rectu.

!. y"* !"x

". x

"

* Ay. y"* 1


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!". Vertices (D A & ,)& foci (D :& ,)

!. >nds of ma;or axis (D & ,)& ends of minor axis (,& D ")

!:. >nds of ma;or axis (,& D 5)& ends of minor axis (D !& ,)

!@. Length of ma;or axis "A& foci (D @& ,)!A. Length of minor axis !A& foci (,& D A).

!9. oci (D & ,)& a * :

!


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6&S"E778!EO%S E(E#"&SE

>.!9. The focus of a parabolic mirror as shown in

ig. is at a distance of @ cm from its vertex.$f the mirror is :@ cm deep& find

>.!


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Type && E+tra Practice $uestions'

!. ind the centre and radius of the circle x"2 y"1 "x 2 :y *


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!


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9. ind the euation of a circle passing through the point (9& ) having radius

units and whose centre lies on the line y * x 1 !.

nds of ma;or axis (,& 5)& end of minor axis (!&,).

::. ind the euation of the ellipse satisfying the condition

( i) /xes along coordinate axes& foci a long x-axis and passing through

(:&) and (1!&:).

(ii) oci at ( &,) and passing through (:&!)

(iii) e * H: & foci on y-axis& centre at origin& passingthrough (A&:).

( iv) Length of minor axis !A& foci (, & A).

:@. / man running a race-course notes that the sum of the distances from the

two flag posts from him is always !, metres and the distance between the

flag post is < metres. ind the euation of the path traced by the man.

:A. /n arch is in the form of a semi-ellipse. $t is


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A,. ind the vertex& focus& directrix and axis of parabolas

(i) (y 1 )"* :a (x 1 )& a = ,

(ii) (x 1 )"* :a (y 1 )& a = ,.

A!. Show that the following euations represent parabolas. $n each case& findvertex& axis& focus& directrix& latus rectum. /lso rough s%etch.

A". Show that the fol lowing euations represent ell ipse. $n each case& f ind

centre& vertices& foci eccentricity& directrices& latus rectum& ma;or axis and

minor axis. /lso draw rough s%etches

(i) :x"2 !Ay"1 ":x 1 "y 1 !" * ,.

(ii) +x"2 :y "1 @:x 1 @Ay 2 ":! * ,.

A. ind the euation of the ellipse whose axes are parallel to the coordinate

axes having its centre at the point ("& 1)& one focus at (& 1) and one

vertex at (:& 1).A:. ind the euation of the ellipse whose foci are (1"&) and ("&) and whose

semi minor axis is 5 .

A@. ind the euation of the hyperbola with foci are (A&:) and (1:&:) and

eccentricity is ".

AA. ind the euation of the hyperbola vertices are at (1


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Type &&& Obective $uestions'

!. The euation of the circle in the f irst uadrant touching each coordinate

axis at a distance of one unit from the origin is (a) 012222 =++ yxyx (b) 012222 =+ yxyx

(c) 02222 =+ yxyx (d) 012222 =++ yxyx

". The euation of the circle having centre (!&1") and passing through the

point of in tersection of the lines x 2 y * !: and " x 2 @y * !< is

(a) 0204222 =++ yxyx (b) 0204222 =+ yxyx

(c) 0204222

=++ yxyx (d) 0204222 =+++ yxyx

. The area of triangle formed by the lines ;oining the vertex of the parabola x"

* !" y to the ends of its latus rectum is

(a) !" s. units. (b) !A s. units(c) !< s. units (d) ": s. units

: . The euations of the lines ;o ining the vertex of the parabola y"* Ax to the

points on it which have abscissa ": are

(a) y "x * , (b) " y x * ,

(c) x "y * , (d) "x y * ,

@. The euation of the ellipse whose centre is at the origin and the x-axis& the

ma;or axis& which passes through the points (1& !) and ("& 1") is

(a) @x"2 y "" (b) x"2 @y "* "

(c) @x"1 y "* " (d) x"2 @y "2 " * ,

A. The length of the transverse axis along x-axis with centre at origin of a

hyperbola is 9 and it passes through the point is 9 and it passes through the

point (@& 1"). The euation of the hyperbola is

(a) 151

196

49

4 22= yx (b) 1

196

51

4

49 22= yx

(c) 1196

51

49

4 22= yx (d) none of these

9. The area of the circle centred at (!&") and passing through (:&A) is

(a) @ (b) !, (c) "@ (d) none of these

uation of a circle which passes though (&A) and touches the axes is

(a) 036622 =++++ yxyx (b) 096622 =+ yxyx

(c) 096622 =++ yxyx (d) none of these

+. >uation of the circle with centre on the y-axis and passing through the

origin and the point ("&) is

(a) 01322 =++ yyx (b) 031333 22 =+++ xyx

(c) 01366 22 =+ xyx (d) 031322 =+++ xyx

!,. The euation of a circle with origin as centre and passing through the

vertices of an euilateral triangle whose median is of length a is


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(a) 222 9ayx =+ (b) 222 16ayx =+

(c) 222 4ayx =+ (d) 222 ayx =+

!!. $ f the focus of a parabola is (, & 1) and its directrix is y * & then i ts

euation is(a) x"* 1!"y (b) x"* !"y

(c) y"* 1!"x (d) y"* !"x

!". $f the parabola y"* :ax passes through the point (&") then the length of its

latus rectum is

(a)3

2(b)

3

4(c)

3

1(d) :

!. $f the vertex of the parabola is point (1& ,) and the directrix is the line x 2

@ * ,& then its euation is

(a) y

"

* < (x 2 ) (b) x

"

* < (y 2 )(c) y"* 1< (x 2 ) (d) y"* < (x 2 @)

!:. The euation of the ellipse whose focus is (!&1!)& the directrix the line x 1 y

1 * , and eccentrici ty2

1is

(a) 071010727 22 =++++ yxyxyx

(b) 07727 22 =+++ yxyx

(c) 071010727 22 =+++ yxyxyx

(d) none

!@. The length of the latus rectum of the ellipse 123 22 =+ yx is

(a) : (b) (c) < (d)3

4

!A. $ f e is the eccentrici ty of the ell ipse 12

2

2

2

=+

b

y

a

x(a b)& then

(a) ( )222 1 eab = (b) ( )222 1 eba =

(c) ( )1222 = eba (d) ( )1222 = eab!9. The eccentric ity of the hyperbola whose latus from rectum is < and

con;ugate axis is eual to half of the distance between the foci is

(a)3

4(b)

3

4(c)

3

2(d) none of these

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(c) 194

22

=yx

(d) none of these

05. State w)et)er t)e stateent are correct or not. 9ustify.

(a) 0ircle on which the coordinates of any point are (" 2 :cos& 1! 2 : sin ) where is a parameter is given by (x 1") "2 (y 2

!) "* !A.

(b) / bar of given length moves with its extremities on two fixed straight

lines at right angles. /ny point of the bar describes an ellipse.

01. State w)et)er t)e stateents are true and false.

a. The line x 2 y * , is a diameter of the circle

x"2 y"2 Ax 2 "y * ,.

b. The shortest distance from the point ("& 19) to the circle

x"

2 y"

1!:x 1 !,y 1!@! * , is eual to @.c . $ f the line lx 2 my * ! is a tangent to the circle x "2 y"* a"& then the point

(l& m) lies on a circle.

d. The point (!&") lies ins ide the circle x"2y "1"x2Ay2!*,.

e . The line lx 2 y 2 n * , wil l touch the parabola y "* :ax if ln * am "

f. $f 8 is a point on the el lipse 12516

22

=+yx

whose foci are S and SJ& then

8S 2 8SJ *


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g. The euation of the circle which passes through the point (:& @) and as its

centre at ("& ") is KKKKKKKKKKKKKKKKKKKKKKKKK.

h. / circle has radius units and its centre lies on the line y * x 1!. $f it

passes through the point (9& )& its euation is KKKKKKKKKKKKKKKKKKKKKKK.i. $f the latus rectum of an ellipse with axis along x-axis and centre at origin is

!,& distance between foci* length of minor axis& then the euation of the

ellipse is KKKKKKKKKKKKKKKKKK.

;. The euation of the parabola whose focus is the point ("&) and directrix is

the line x 1 :y 2 * , is KKKKKKKKKKKKKKKKKKK.

%. The eccentricity of the hyperbola 12

2

2

2

=

b

y

a

x which passes through the

points (& ,) and 2,23 is KKKKKKKKKKKKKKKKKKKKKK.


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Type


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E+ercise 11.4


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6iscellaneous E+ercise on ")apter < 11

Type && 8nswers E+tra practice $uestions'

!. (!&1") & 13 ". y * " &

circul and conic section (2)

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