Ellipse

ME – 1

ELLIPSE C1

Definitions It is a locus of a point which moves in such a way that the ratio of its distance from a fixed point (called focus) and a fixed line (called directrix) is constant which is less than one. The constant ratio is called eccentricity (e).

C2

Standard Equation Standard equation of an ellipse is

x2 y 2   1 , where a > b and b2 = a2(1 – e2). a2 b2

b2 Eccentricity : e  1  2 , ( 0  e  1) a Foci : S (a e, 0) and

S (– a e, 0).

Equations of Directrices : x 

a a and x   . e e

Major Axis : The line segment AA in which the focii S and S lie (of length 2a) is called the major axis of the ellipse. Point of intersection of major axis with directrix is called the foot of the directrix (z and z  ). Minor Axis : The y-axis intersects the ellipse in the points B (0,  b ) and B (0, b ) . The line segment B B is of length 2b is called the minor axis of the ellipse. Principal Axis : The major and minor axes together are called principal axis of the ellipse. Vertices : Point of intersection of ellipse with major axis A (–a, 0) and A (a, 0) are called vertices. Focal Chord : A chord which passes through a focus is called a focal chord. Double Ordinate : A chord perpendicular to the major axis is called a double ordinate. Latus Rectum : The focal chord perpendicular to the major axis is called the latus rectum. Length of latus rectum (LL) 

2b 2 (minor axis)2   2a(1  e 2 ) a major axis

Centre : The point which bisects every chord of the conic drawn through it, is called the centre of the conic.

x2 y 2 The point C (0, 0) is the centre of the ellipse 2  2  1 . a b Parametric Representation : The equations x = a cos  and y = b sin  together represent the ellipse

x2 y 2   1 where  is a parameter.. a2 b2 Position of a Point w.r.t. an Ellipse : The point P(x1, y1) lies outside, inside or on the ellipse according as

x12 y 12   1   or  0 a2 b2

Auxilliary Circle / Eccentric Angle : A circle described on major axis of ellipse as diameter is called the auxiliary circle.

Einstein Classes,

Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi – 110 018, Ph. : 9312629035, 8527112111

ME – 2

Let Q be a point on the auxiliary circle x2 + y2 = a2 such that line through Q perpendicular to the x-axis on the way intersects the ellipse at P, then P and Q are called as the Corresponding Points on the ellipse and the auxiliary circle respectively. ‘’ (–  <   ) is called the Eccentric Angle of the point P on the ellipse The coordinate of P() and Q() are given by (a cos , b sin ) and (a cos , a sin ) respectively. Also

l ( PN ) b Semi min or axis   l (QN ) a Semi major axis If from each point of a circle perpendicular are drawn upon a fixed diameter then the locus of the points dividing these perpendiculars in a given ratio is an ellipse of which the given circle is the auxiliary circle. Important Points : (i)

If the equation of the ellipse is given as

x2 y 2   1 and nothing is mentioned then the rule a2 b2

is to assume that a > b. (ii)

If b > a is given, then the y-axis will become major axis and x-axis will become the minor axis and all other points and lines will change accordingly.

Practice Problems : 1.

Find the lengths of the major and minor axes; coordinates of the vertices and the foci, the eccentricity and length of the latus rectum of the ellipse

x2 y 2   1. 16 9

2.

Find the length of the major and minor axes; coordinates of the vertices and the foci; the eccentricity and length of the latus rectum of the ellipse : 4x2 + 9y2 = 144 .

3.

Find the lengths of the major and minor axes; coordinates of the vertices and the foci; the eccentricity and length of the latus rectum of the ellipse

x2 y 2   1. 4 36

4.

Find the lengths of the major and minor axes; coordinates of the vertices and the foci; the eccentricity and length of the latus rectum of the ellipse : 4x2 + y2 = 100.

5.

Find the equation of an ellipse whose vertices are at (± 5, 0) and foci at (± 4, 0).

6.

Find the equation of an ellipse whose foci are (± 4, 0) the eccentricity is

7.

Find the equation of an ellipse whose major axis lie on the x-axis and which passes through the points (4, 3) and (6, 2).

8.

Find the equation of the ellipse, the ends of whose major axis are (± 3, 0) and the ends of whose minor axis are (0, ± 2).

9.

Find the equation of the ellipse whose centre lies at the origin axis lies on the x-axis, the eccentricity is

1 . 3

2 and the length of the latus rectum is 5 units. 3

Einstein Classes,


ME – 3 10.

Find the equation of an ellipse whose vertices are (0, ± 13) and the foci are (0, ± 5).

11.

Find the equation of the ellipse whose foci are (0, ± 6) and the length of whose minor axis is 16.

12.

Find the equation of the ellipse whose foci are (0, ± 5) and the length of whose major axis is 20.

13.

Find the equation of the ellipse for which e 

14.

Find the equation of the ellipse with centre at the origin, major-axis on the y-axis and passing through the points (3, 2) and (1, 6).

15.

Find the lengths and equations of the focal radii drawn from the point (43, 5) on the ellipse 25x2 + 16x2 = 1600.

16.

Find the position of the point (4, –3) relative to the ellipse 25x2 + 16x2 = 1600.

17.

Find the equation of an ellipse whose focus is (–1, 1), eccentricity is

4 and whose vertices are (0, ± 10). 5

1 and the directrix is 2

x – y + 3 = 0. 18.

Show that the line lx + my + n = 0 will cut the ellipse x2/a2 + y2/b2 = 1 in points whose eccentric angles differ by (/2) if a2l2 + b2m2 = 2n2.

19.

If the angle between the straight lines joining foci and the ends of the minor axis of the ellipse

x2 a2 20.



y2 b2

 1 is 900. Find its eccentricity..

The extremities of a line segment of length l move in two fixed perpendicular straight lines. Find the locus of that point which divides this line segment in ratio 1 : 2.

[Answers : (5)

x2 y 2 x2 y2 x2 y 2 x2 y 2   1 (6)   1 (7)   1 (9) 20x2 + 36y2 = 405   1 (8) 25 9 144 128 9 4 52 13

x2 y 2 x2 y 2 x2 y 2 x2 y 2 x2 y2   1 (12)   1 (13)   1 (14)  1   1 (11) 64 100 75 100 36 100 10 40 144 169 (15) 7, 13, x + 43y – 243 = 0, 11x –43y – 243 = 0 (17) 7x2 + 7y2 + 2xy + 10x – 10y + 7 = 0 (19) e = 1/2 (20) 9x2 + 36y2 = 4l2 ] (10)

C2

Line and an Ellipse : The line y = mx + c meets the ellipse

x2 y 2   1 in two points real, coincident or imaginary according as a2 b2

c2 is < = or > a2m2 + b2. Hence y = mx + c is tangent to the ellipse

x2 y 2   1 if c2 = a2m2 + b2. a2 b2

Practice Problems :

x2

y2

 1 if a2l2 + b2m2 = n2.

1.

Prove that the straight line lx + my + n = 0 touches the ellipse

2.

For what value of  does the line y = x +  touches the ellipse 9x2 + 16y2 = 144.

a2



b2

[Answers : (2)  = ± 5] C3

Different forms of Tangents : (a)

Slope form : y  mx  a 2m 2  b 2 is tangent to the ellipse

x2 y 2   1 for all values of a2 b2

m.

Einstein Classes,


ME – 4 xx1 yy 1 x2 y 2  2  1 is tangent to the ellipse 2  2  1 at (x1, y1) 2 a b a b

(b)

Point form :

(c)

Parametric form :

x2 y 2 x cos  y sin    1 is tangent to the ellipse 2  2  1 at the point a b a b

(a cos , b sin ). Important Points : (i)

There are two tangents to the ellipse having the same m, i.e. there are two tangents parallel to any given direction. These tangents touches the ellipse at extremities of a diameter.

(ii)

   sin  cos 2 2 Point of intersection of the tangents at the point  and  is  a ,b    cos  cos 2 2 

(iii)

The eccentric angles of the points of contact of two parallel tangents differ by .

  .   


Find the equation of the tangents to the ellipse 3x2 + 4y2 = 12 which are perpendicular to the line y + 2x = 4.

2.

Prove that the locus of mid-points of the portion of the tangents to the ellipse

x2 a2



y2 b2

 1 intercepts

between the axes is a2y2 + b2x2 = 4x2y2. [Answers : (1) x – 2y ± 4 = 0] C4

Different forms of Normals :

x2 y 2 a2x b2y   1 is   a2  b2 . a2 b2 x1 y1

(i)

Equation of the normal at (x1, y1) to the ellipse

(ii)

Equation of the normal at the point (a cos  . b sin ) to the ellipse

x2 y 2   1 is; a2 b2

ax sec – by cosec  = (a2 – b2). (iii)

Equation of a normal in terms of its slope ‘m’ is y  mx 

(a 2  b 2 )m a 2  b 2m 2

.

In general, four normals can be drawn to an ellipse from any point and if , , ,  the eccentric angles of these four co-normal points then  +  +  +  is an odd multiple of . If , ,  are the eccentric angles of three points on the ellipse

x2 a2



y2 b2

 1 , the normals at which are

concurrent, then sin( + ) + sin( + ) + sin( + ) = 0. Practice Problems : 1.

If the normal at an end of a latus-rectum of an ellipse

x2 a2



y2 b2

 1 passes through one extremity of

the minor axis, find the eccentricity of the ellipse.

Einstein Classes,


ME – 5

2.

A normal inclined at an angle of 450 to x-axis of the ellipse

y2

 1 is drawn. It meets the major a2 b2 and minor axis in P and Q respectively. If C is the centre of the ellipse, prove that area of CPQ is

(a 2  b 2 ) 2 2(a 2  b 2 ) 3.

x2



sq. units.

Find the equation of the curve on which co-normal points will lie of the ellipse

x2 a

2



y2 b2

 1 if normals

are drawn from the point (h, k).

5 1 (3) (a2 – b2)xy + b2kx – a2hy = 0] 2

[Answers : (1)

C5

Pair of Tangents : The equation to the pair of tangents which can be drawn from any point (x1, y1) to the ellipse

is given by SS1 = T2 where S  C6

x2 y 2  1 a2 b2

x12 y 12 x2 y 2 xx yy S   2  1 ; T  21  21  1   1 ; 1 2 2 2 a b a b a b

Director Circle : Locus of the point of intersection of the tangents which meet at right angles is called the Director Circle. The equation to this locus is x2 + y2 = a2 + b2 i.e. a circle whose centre is the centre of the ellipse and whose radius is the length of the line joining the ends of the major and minor axes.

C7

Chord of Contact : Equation to the chord of contact of tangents drawn from a point P(x1, y1) to the ellipse T = 0 where T 

C8

x2 y 2   1 is a2 b2

xx1 yy 1  2 1. a2 b

Chord with a given middle point : Equation of the chord of the ellipse

S1 

x2 y 2   1 whose middle point is (x1, y1) is T = S1 where a2 b2

xx yy x12 y 12  2  1 and T  21  21  1 2 a b a b


Find the locus of the points of the intersection of tangents to ellipse

x2 y 2   1 which make an a2 b2

angle . 2.

Prove that the locus of the middle points of normal chords of the ellipse

 x2 y 2      a2 b2   

2

x2 y 2   1 is the curve a2 b2

 a6 b 6     (a 2  b 2 ) 2 .   x2 y 2   

Einstein Classes,


ME – 6

3.

Tangents at right angles are drawn to the ellipse

x2 a2



y2 b2

 1 . Show that the locus of the middle

2

 x2 y 2  x2  y 2 points of the chord of contact is the curve  2  2   2 . a b  a  b2  [Answers : (1) (x2 + y2 – a2 – b2)2 tan2 = 4(b2x2 + a2y2 – a2b2)] C9

Diameter : The locus of the middle points of a system of parallel chords is called a diameter and the point where the diameter intersects the ellipse is called the vertex of the diameter. Two diameters are said to be conjugate when each bisects all chords parallel to the other. Properties of Conjugate Diameter Prop.1 : The eccentric angles of the ends of a pair of conjugate diameters of an ellipse differ by a right angle. Prop. 2 : The sum of the squares of any two conjugate semi diameters of an ellipse is constant and equal to the sum of the squares of the semi-axes of the ellipse i.e., CP2 + CD2 = a2 + b2. Prop. 3 : The product of the focal distances of a point on an ellipse is equal to the square of the semi diameter which is conjugate to the diameter through the point. Prop. 4 : The tangents at the extremities of a pair of conjugate diameters from a parallelogram whose area is constant and equal to product of the axes. Practice Problems :

1.

Find the equation of the diameter bisecting chords of slope m of the ellipse

2.

If y = mx and y = m1x be two conjugate diameters of an ellipse

mm 1  3.

b2 a2

a2



a2 y2 b2



y2 b2

 1.

 1 then prove that

.

Show that the tangents at the ends of conjugate diameters of the ellipse x2/a2 + y2/b2 = 1 intersect on the ellipse x2/a2 + y2/b2 = 2. [Answers : (1) y  

C10

x2

x2

b 2x a 2m

]

Pole and Polar From any point P (inside or outside the ellipse), a straight line is drawn which intersects the ellipse at Q and R. Now the tangents are drawn to the ellipse at Q and R. The locus of the point of intersection of these tangents is called the polar of the given point P with respect to ellipse and the point P is called the pole of the polar. Practice Problems :

1.

From any point P(x1, y1) (inside or outside the ellipse), a straight line is drawn which intersects the ellipse

x2 y 2   1 at Q and R. Now the tangents are drawn to the ellipse at Q and R. Find the locus a2 b2

of the point of intersection of these tangents ? 2.

Find the pole of the given line px + qy + r = 0 with respect to the ellipse

Einstein Classes,

x2 y 2  1 ? a2 b2


ME – 7 3.

The perpendicular from the centre of an ellipse x2/a2 + y2/b2 = 1 on the polar of a point with respect to the ellipse is constant and equal to c. Prove that the locus of the point is the ellipse

x2 a

[Answers : (1)

C11

xx1 a2



yy 1 b2

4



y2 b

4



1 c2

.

  a 2p b 2q  ] ,  1 (2)  r   r

Reflection theory of elliptical reflector : Consider an elliptical reflector whose foci are the points S1 and S2. All rays emanating from the point S1 will pass through the point S2 after undergoing a reflection.


A ray emanating from the point (–3, 0) is incident on the ellipse 16x2 + 25y2 = 400 at the point P with ordinates 4. Find the equation of the reflected ray after first reflection. [Answers : (1) 4x + 3y = 12]

C12

Concylic Points : Any circle intersects an ellipse in two or four real points. They are called concyclic points and the sum of their eccentric angles is an even multiple of . Practice Problems :

1.

Prove that the common chords of an circle and an ellipse are equally inclined to the axes of the ellipse.

2.

P CP and DCD are conjugate diameters of an ellipse and  is the eccentric angle of P. Prove that

the eccentric angle of the point where the circle through P, P , D again cuts the ellipse is

C13

  3 . 2

Important Points : Refering to the ellipse

x2 y 2   1. a2 b2

S as its foci then l(SP) + l( S P) = 2a.



If P be any point on the ellipse with S and



The tangent and normal at a point P on the ellipse bisect the external and internal angles between the focal distance of P. This refers to the well known reflection property of the ellipse which states that rays from one focus are reflected through other focus and vice-versa. Hence we can deduce that the straight lines joining each focus to the foot of the perpendicular from the other focus upon the tangents at any point P meet on the normal PG and bisects it where G is the point where normal at P meets the major axis.



The product of the length’s of the perpendicular segment from the foci on any tangent to the ellipse is b2 and the feet of these perpendiculars lie on its auxiliary circle and the tangents at these feet to the auxiliary circle meet on the ordinates of P and that their point of intersection is a similar ellipse as that of the original one.



The portion at any point P on the ellipse with centre C meet the major and minor axes in G and g respectively and if CF be perpendicular upon this normal then (i)

PF . PG = b2

(ii)

PF . Pg = a2

Einstein Classes,


ME – 8

S P

(iii)

PG . Pg = SP .

(iv)

CG . CT = CS2

(v)

locus of the mid point of Gg is another ellipse having the same eccentricity as that of the original ellipse.

S are the focii of the ellipse and T is the point where tangent at P meet the major axis]

[where S and 

The circle of any focal distance as diameter thoches the auxiliary circle. Perpendicular from the centre upon all chords which join the ends of any perpendicular diameters of the ellipse are the constant length.



If the tangent at the point P of a standard ellipse meets the axis in T and t and CY is the perpendicular on it from the centre then, (i)

T t . P Y = a2 – b2 and

(ii)

least value of T t is a + b

SINGLE CORRECT CHOICE TYPE 1.

2.

3.

On the ellipse 4x2 + 9y2 = 1, the points at which the tangents are parallel to the line 8x = 9y are (a)

 2 1  ,   5 5

(c)

 2 1   ,   5 5

5.

 2 1  ,   5 5

point P

(d)

 2 1  ,   5 5

tangent, nearer to the point P to the circle x2 + y 2 = 1 and the hyperbola x2 – y 2 = 1. The equation of the ellipse in standard form is

 1  . Its one directrix is the common , 1 2 

If P(x, y), F 1 = (3, 0), F 2 = (–3, 0) and 16x2 + 25y2 = 400, then PF1 + PF2 equals (a)

8

(b)

6

(c)

10

(d)

12

2

(a)

1  9 x    12( y  1) 2  1 3 

(b)

1  9 x    12( y  1) 2  1 3 

(c)

1  9 x    12( y  1) 2  1 3 

(d)

1  9 x    12( y  1) 2  1 3 

2

The radius of the circle passing through the foci of

2

is (a)

4

(b)

3

(c)

12

(d)

7/2

2

Let E be the ellipse

2

2

x y   1 and C be the circle 9 4

x2 + y2 = 9. Let P and Q be the points (1, 2) and (2, 1) respectively. Then : (a)

Q lies inside C but outside E

(b)

Q lies outside both C and E

(c)

P lies inside both C and E

(d)

P lies inside C but outside E

Einstein Classes,

1 and one focus at the 2

(b)

x 2 y2   1 and having its centre (0, 3) the ellipse 16 9

4.

An ellipse has ecentricity

6.

The area of the rectangle formed by the perpendiculars from the centre of the ellipse to the tangents and normals at the point whose ecentric angle

 is 4

(a)

 a 2  b2   2 ab (b) 2  a  b  

(c)

1  a 2  b2   ab ab  a 2  b 2 

 a 2  b2   2 ab 2  a  b  


ME – 9

(d) 7.

12.

1  a 2  b2   ab ab  a 2  b 2 

The tangent at a point P (a cos , b sin ) on the

If  and  are the ecentric angles of the extremities of a focal chord of an ellipse, then the ecentricity of the ellipse is (a)

cos   cos  (b) cos(  )

sin   sin  sin(  )

(c)

cos   cos  (d) cos(  )

sin   sin  sin(  )

x 2 y2 ellipse 2  2  1 meets the auxiliary circle in a b two points. The chord joining them subtends a right angle at the centre. Then the eccentricity of the ellipse is given by :

8.

(a)

(1 + sin2)–1/2

(b)

(1 + cos2)–1/2

(c)

(1 + sin2)1/2

(d)

(1 + cos2)1/2

P is a variable point on the ellipse

13.

x 2 y2   1 meets the axis in G and g a 2 b2

x 2 y2   1 with a 2 b2

respectively, then P G : P g =

A A  as the major axis, then the maximum value of the area of the triangle A P A  is

9.

(a)

ab

(b)

(c)

ab/2

(d)

14.

2ab none of these

a:b

(b)

a2 : b2

(c)

b2 : a2

(d)

b:a

The equation

x2 y2   1 if (r > 1) represents 1 r 1 r

An ellipse

The slope of a common tangent to the ellipse

(b)

A hyperbola

x 2 y2   1 and a concentric circle of radius r is a 2 b2

(c)

A circle

(d)

An imaginary ellipse

15. (a)

(c)

tan 1

2

r b (b) a2  r2

 r 2  b2   2 2  a r 

(d)

2

2

r b a2  r2 a2  r2 r 2  b2

The equation of the tangent to the ellipse 4x2 + 3y2 = 5 which are parallel to the line y = 3x + 7 are

y  3x 

(b)

155 y  3x  12

(d)

y  3x 

16.

155 3

(a)

(c)

11.

(a)

(a)

2

10.

If the normal at any point P on the ellipse

(a)

y = 3, x + y = 5

(b)

y = –3, x – y = 5

(c)

y = 4, x + y = 3

(d)

y = –4, x – y = 3

If any tangent to the ellipse

h2

17.

18.

none of these

The line x = at2, meets the ellipse

x 2 y2   1 in a 2 b2

real points iff (a)

|t| < 2

(b)

(c)

|t|  1

(d)

x2

y2

 1 cuts off a2 b2 intercepts of length h and k on the axes then a2

95 12

Einstein Classes,

The equation of the tangents of the ellipse 9x2 + 16y2 = 144 which passes through the point (2, 3) is



b2 k2





(a)

0

(b)

1

(c)

–1

(d)

none

Eccentric angle of a point on the ellipse x2 + 3y2 = 6 at a distance 2 units from the centre of the ellipse is (a)

450

(b)

900

(c)

600

(d)

300

The distance of the centre of the ellipse x2 + 2y2 – 2 = 0 to those tangents of the ellipse which are equally inclined from both the axes is (a)

2 3

(b)

3 2

(c)

3

(d)

2

|t| > 1 none of these


ME – 10 19.

20.

21.

On the ellipse 4x2 + 9y2 = 1, the points at which the tangents are parallel to the line 8x = 9y are (a)

 2 1  ,   5 5

(b)

 5    ,  5 2  

(c)

 1 5  ,   5 2

(d)

none of these

23.

ellipse

(b)

hyperbola

(c)

parabola

(d)

circle

If 3 bx + ay = 2ab touches the ellipse

x2

of  such that the sum of intercepts on axes made by this tangent is minimum is

a2

24.



y2 b2

22.

0

(c)

 6

1

 3

(b)

 6

(c)

 8

(d)

 4

If  and  are ecentric angles of the ends of a

tan

(b)

 3

(d)

 2

The eccentric angle of a point on the ellipse 2

(a)

x 2 y2 focal chord of the ellipse 2  2  1 , then a b

then the eccentric angle  is (a)

  tan is equal to 2 2

(a)

1 e 1 e

(b)

(c)

e 1 e 1

(d)

2

x y   1 whose distance from the centre of the 6 2 ellipse is 2 is (a)

0

(c)

 4

(b)

 6

(d)

 2

x2  y 2  1 at 27

  (33 cos , sin ) where    0,  . Then the value  2

If a variable tangent of the circle x 2 + y 2 = 1 intersects the ellipse x2 + 2y2 = 4 at points P and Q, then the locus of intersection of tangents at P and Q is (a)

Tangent is drawn to ellipse

25.

If chords of the ellipse

x2

e 1 e 1 none

y2

 1 pass through a a2 b2 fixed point (h, k) then the locus of their middle points is a 

(a)

parabola

(b)

ellipse

(c)

hyperbola

(d)

none of these

ANSWERS (SINGLE CORRECT CHOICE TYPE)

Einstein Classes,

1.

b

10.

b

19.

a

2.

c

11.

c

20.

a

3.

a

12.

d

21.

c

4.

d

13.

c

22.

c

5.

a

14.

d

23.

b

6.

a

15.

a

24.

b

7.

a

16.

b

25.

b

8.

a

17.

c

9.

a

18.

b


ME – 11

EXCERCISE BASED ON NEW PATTERN COMPREHENSTION TYPE

5.

The centre of the ellipse will be

Comprehension-1 The equation of the ellipse is given

(a)

 1  1,   3

(b)

1   , 1 3 

(c)

1    1,  3 

(d)

1   1,   3 

4(x – 2y + 1)2 + 9(2x + y + 2)2 = 25 1.

2.

The centre of the ellipse will be (a)

(0, 1)

(b)

(0, –1)

(c)

(–1, 0)

(d)

(1, 0)

(a)

(c) 3.

6.

The vertex of the ellipse will be

The vertex of the ellipse will be 3   ,1 2 

 1    ,  1 2  

(a)

(1, 0)

(b)

(b)

 1    , 1  2 

 2  1,   3

(c)

 2  0,   3

(d)

none of these

(d)

1   ,  1 2 

Comprehension-3

The focus of the ellipse will be

(a)

Two diameters are said to be conjugate when each bisects all parallel chords. If y = m1x and y = m2x be the two conjugate diameters of an ellipse

   5  6 , 5   6 3  

x2

(b)

   5  6 , 5   6 3  

(c)

   5  6 , 5   6 3  

(d)

   5  6 , 5   6 3  

7.

Comprehension-2 An ellipse has the eccentricity

1 and one focus is 2

8.

1  F , 1  . The corresponding directrix is the 2 

common tangent, nearer to F, to the circle x2 + y2 = 1 and the hyperbola x2 – y2 = 1, then 4.

Equation of the ellipse will be (a)

3x2 + 4y2 – 2x – 8y + 4 = 0

(b)

3x2 + 4y2 – 2x + 8y + 4 = 0

(c)

3x2 + 4y2 + 2x – 8y + 4 = 0

(d)

2

2

3x + 4y – 2x – 8y – 4 = 0

Einstein Classes,

y2

9.



2

 1 then m 1m 2 

b2

. Conjugate diama b a2 eters of circle are perpendicular to each other hence the angle between conjugate diameters of ellipse > 900. 2

The eccentric angles of the ends of a pair of conjugate diameters of an ellipse is differ by (a)

0

(b)

 6

(c)

 3

(d)

 2

The sum of the squares of any two conjugate semi diameters of an ellipse is constant then (a)

(CP ) 2  (CD ) 2  a 2  b 2

(b)

(CP ) 2  (CD ) 2  a 2  b 2

(c)

(CP) 2  (CD) 2  2a

(d)

(CP ) 2  (CD ) 2  2b

The length of the equi-conjuate diameter will be (a)

a2  b2 2

(b)

a2  b2

(c)

2 a2  b2

(d)

equi conjugate diameter not exist


ME – 12 MATRIX-MATCH TYPE

(C)

Matching-1 Column - A

/2

(B)

The angle between the tangents from the point (2, 2) to the ellipse 4x2 + 9y2 = 36

0

(C)

The inclination to the (r) major axis of the diameter

(q)

4 cuts the x 3 and y-axis at the points A and B respectively. If O is the origin then area of triangle OAB is equal to (D)

/4

x2 y2  1 625 4096 whose length is the GM between the major and minor axes, is If the tangent to the ellipse x2 + 4y2 = 16 at the point P  is

(s)

tan–18/5

1.

Column - A

Column - B

Consider an ellipse

(p)

(B)



16 

The distance of the any point present on the ellipse

(a)



 2

(b)



3 2

(c)



5 2

(d)



7 2

y2

 1 , centered a b2 at point ‘O’ and having AB and CD as it’s major and minor axes respectively. If S1 be one of the focus of the ellipse, radius of incircle of triangle OCS1 be 1 unit and OS1 = 6 units, then area of the ellipse is equal to 2

The maximum distance of the center of the

Einstein Classes,

2.

3. (q)

x2 y 2   1 from 9 4 the chord of contact of mutually perpendicular tangents of the ellipse is ellipse

11/15

x2 y 2   1 from the centre of the ellipse is 2. Then 6 2 the eccentric angle  will be

Matching-2

x2

Consider an ellipse and (s) a concentric circle. The circle passes through the foci of the ellipse and intersects the ellipse in four distinct points. The length of major axis of the ellipse is 15 units. If S1 and S2 are the foci of the ellipse, P be one of point of intersection of ellipse and circle and area of triangle PS1S2 is 26 sq. units, then eccentricity of the ellipse is equal to

MULTIPLE CORRECT CHOICE TYPE

a normal to the circle x2 + y2 – 8x – 4y = 0 then  is equal to

(A)

24

slope 

of the ellipse

(D)

(r)

x2 y 2   1 , having 18 32

Column - B

The inclination to the (p) major axis of the diameter of the ellipse, the square of whose length is the harmonic mean between the squares of the major and minor axes, is

(A)

A tangent to the ellipse

9/3

A latus rectum of an ellipse is the line which is (a)

passing through the focus

(b)

perpendicular to major axis

(c)

parallel to minor axis

(d)

parallel to tangent at vertex

x2 y 2  1 9 4 which cut off equal intercept on the axis is The equation of tangent to the ellipse

(a)

y = x + 13

(b)

y = – x + 13

(c)

y = – x – 13

(d)

y = x – 13


ME – 13 4.

If the normal at any point P of the ellipse

x2

STATEMENT-1 : The sum of the squares of the reciprocals of two perpendicular diameters of an ellipse is constant. STATEMENT-2 : The product of perpendiculars drawn from the foci on any tangent to an ellipse is constant.

3.

STATEMENT-1 : The equation

y2

 1 meets the major and minor axes at G a2 b2 and E respectively, and if CF is perpendicular upon this normal from the centre C of the ellipse, then

5.

1.



2.

(a)

PF.PG = b2

(b)

PF.PE = a2

(c)

PF.PG = a2

(d)

PF.PE = b2

The equation of normals at the point of intersection of the line 2x + y = 3 and ellipse 4x2 + y2 = 5 are (a) 2x – 2y + 3 = 0 (b) x – 4y + 3 = 0 (c) 2x + 2y + 3 = 0 (d) x – 4y – 3 = 0 Assertion-Reason Type Each question contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason). Each question has 4 choices (A), (B), (C) and (D) out of which ONLY ONE is correct. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1 (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True The curve C is a locus of a point whose sum of distance from the point (3, 0) and (–3, 0) is 4. STATEMENT-1 : The curve C cuts-off intercept two unit from the line 2y – 1 = 0. STATEMENT-2 : The curve C represents the equation

4.

x2 y2  1 6a a2 will represent an ellipse for a  (2, 6) ~ {4}. STATEMENT-2 : The eccentricity of the ellipse is less than 1. STATEMENT-1 : P is any variable point on the

5.

x2

x2 a2



y2 b2



y2

 1 having the points S1 and S2 as a2 b2 it’s foci. Maximum area of the triangle PS1S2 is equal to abe, where e is the accentricity of the ellipse. STATEMENT-2 : The locus of incentre of the triangle will be an ellipse. STATEMENT-1 : Let  1 ,  2 ,  3 ,  4 are the eccentric angles of four points on an ellipse ellipse

 1 , then 1 + 2 + 3 + 4 = 2n, n  N.

STATEMENT-2 : These points are concyclic.

x2  y2  1 . 4

(Answers) EXCERCISE BASED ON NEW PATTERN COMPREHENSION TYPE 1.

c

2.

c

3.

b

4.

a

7.

d

8.

a

9.

b

2.

[A-p; B-q; C-r; D-s]

5.

b

6.

4.

a, b

d

MATRIX-MATCH TYPE 1.

[A-r; B-s; C-s; D-p, q]

MULTIPLE CORRECT CHOICE TYPE 1.

a, b, c, d

5.

a, b

2.

a, b, c, d

3.

a, b, c, d

4.

B

ASSERTION-REASON TYPE 1.

D

2.

Einstein Classes,

B

3.

A

5.

A


ME – 14

INITIAL STEP EXERCISE (SUBJECTIVE) 1.

2.

x 2 y2   1 a chord through the If in the ellipse 36 9 point (4, 0) has the point (2, ) as its midpoint, find . Find the locus of mid point of the chord of the ellipse

3.

4.

5.

6.

at the centre. Find the centre, the lengths of axes, eccentricity, foci, directrices and length of latus rectum of the ellipse 2x2 + 3y2 – 4x – 12y + 13 = 0. Find the centre, foci, length of latus rectum, directrices of the following ellipse. x2 + 5y2 + 2xy – 2x – 18y + 13 = 0 Find the equation of an ellipse whose foci are (2, 3) and (–2, 3) and whose semi minor axis is of length 5. Find the locus of the foot of perpendicular drawn from (0, 0) to any tangent to the ellipse

8.

9.

10.

tangent to the ellipse 11.

12.

14.

15.

x 2 y2 the curve 2  2  4 . a b

Einstein Classes,

tx y x ty   t  0 and   1  0 . Prove a b a b

that they intersect on an ellipse at a point whose eccentric angle is 2 tan–1t. Find the locus of a point P, tangents from which to

x 2 y2 the ellipse 2  2  1 make angles 1 & 2 with a b

16.

18.

the major axis ; find the locus when (a) 1 + 2 = 2 (b) tan1 + tan2 = c (c) tan1 – tan2 = d (d) tan21 + tan22 =  Find the locus of the intersection of tangents (a) which meet at an angle . (b) If the sum of eccentric angles of their points of contact be equal to 2 Prove that (a) If SY an S’Y’ be the perpendiculars from the foci upon the tangent at any point P of the ellipse, then Y and Y’ lie on the auxiliary circle, and SY . S’Y’ = b2 (b) If the normal at any point P meet the major and minor axes in G and g, and if CF be the perpendicular upon this normal, then PF . PG = b2 & PF . Pg = a2. The eccentric angles of the extremities of a chord

x 2 y2 of an ellipse 2  2  1 are 1 and 2. The chord a b cuts the major axis of the ellipse at a distance d from the centre. Prove that

x 2 y2   1. a 2 b2

Prove that the perpendicular from the focus upon any tangent and the line joining the centre to the point of contact meet on the corresponding directrix. Prove that the locus of the middle points of the portions of tangents included between the axes is

a 2  b 2 from the centre is 2a2.

Find the locus of the point of intersection of the lines

17.

If the normal at an end of a latus rectum of an ellipse passes through one extremity of the minor axis, show that the eccentricity of the curve is given by e4 + e2 – 1 = 0. Q is the point on the auxiliary circle corresponding to P on the ellipse, PLM is drawn parallel to CQ to meet the axes in L & M; prove that PL = 6 and PM = a. Prove that the extremities of the latus rectum of all ellipses, having a given major axis 2a, lie on the parabola x 2 = – a(y – a), or on the parabola x2 = a(y + a). Find the condition that the line lx + my = n be a

Prove that the sum of squares of the perpendiculars on any tangent from two points on the minor axis, each distant

x 2 y2   1 which subtends a right angle a 2 b2

x 2 y2   1. a 2 b2 7.

13.

tan 19.

1  da tan 2  . 2 2 da

Prove that the line lx + my = n is a normal to the 2

a 2 b2  a 2  b2   . ellipse if 2  2   l m  n  20.

Tangents are drawn from a point P to the circle x 2 + y 2 = r 2 so that the chords of contact are tangents to the ellipse a2x2 + b2y2 = r2. Find the locus of P.


ME – 15 21.

Find the locus of middle points of normal chords of

x 2 y2 the ellipse 2  2  1 . a b

FINAL STEP EXERCISE (SUBJECTIVE) 1.

Let a tangent to the ellipse

x 2 y2   1 meets the a 2 b2

major and minor axis in P & Q respectively. Let R be a point which divides PQ in the ratio 2 : 1. Find the locus of R. 2.

ab the line touches a circle of radius concentric with the ellipse. 10.

x 2 y2   1 at the points a 2 b2

drawn through the positive end of minor axis is

x 2 y2 y   . a 2 b2 b

 whose eccentric angles differ by . 2 3.

The tangent and normal to the ellipse x2 + 4y2 = 4 at a point P on it meet the major axis in Q & R respectively. If QR = 2, show that the eccentric angle 1

11.

5.

6.

7.

8.

9.

 &  are eccentric angle of two variable points P

x 2 y2 and Q on the ellipse 2  2  1 . If  +  = 2, a b

2 . 3

prove that PQ is always parallel to the tangents at  and conversely.

of P is cos  4.

Show that the locus of the middle points of those

x 2 y2 y chords of the ellipse 2  2  . Which aree a b b

Find the locus of the point of intersection of tangents drawn to ellipse

Any point P of an ellipse is joined to the extremities of the major axis. Prove that the portion of directrix intercepted by them subtend one right angle at the ‘corresponding’ focus.

12.

Any tangent to an ellipse is cut by the tangents at the major axis in T, T’. Prove that the circle with diameter TT’ will pass through the foci.

13.

Let d be the perpendicular distance from the

From any point P on the ellipse PN is drawn perpendicular to the axis and produced to Q, so that NQ equals PS, where S is a focus. Prove that the locus of Q is the two straight lines y ± ex + a = 0.

centre of the ellipse

Prove that the sum of the eccentric angles of the extremities of a chord, which is drawn in a given direction, is constant, and equal to twice the eccentric angle of the point at which the tangent is parallel to the given direction. If the tangent drawn at a point (t2, 2t) on the parabola y2 = 4x is the same as the normal drawn at a point (5 cos , 2 sin ) on the ellipse 4x2 + 5y2 = 20, find the values of t and .

x 2 y2   1 to the tangent a 2 b2

drawn at a point P on ellipse given. If F1 & F2 are two foci of the ellipse, then show that

Prove that the straight lines, joining each focus to the foot of the perpendicular from the other focus upon the tangent at any point P, meet on the normal at P & bisect it.

(PF1 – PF2)2 = 4a2 14.

 b2  1  2  .  d 

If p is the length of perpendicular from the focus s

x 2 y2 of the ellipse 2  2  1 on the tangent at P then a b show that 15.

b 2 2a   1. P 2 SP

Q is a point on the auxiliary circle corresponding

If the portion of the line x cos  + y sin  = c 2

intercepted by the ellipse

2

x y  2  1 subtends a 2 a b

right angle at the centre of the ellipse. Prove that

Einstein Classes,

a 2  b2

to a point P on the ellipse

x 2 y2   1 . Two points a 2 b2

L and M are selected on the x-axis and y-axis respectively such that the line PLM is || to OQ. Prove that PL : PM = b : a.


ME – 16 16.

(a)

Show that the angle between the tangents

x 2 y2 to the ellipse 2  2  1 from a point and also a b

circle x2 + y2 = ab at their point of

that the sum of ecentric angles of their feet is equal to an odd multiple of two right angles.

tan 1

ab . ab

21.

A circle of radius r, is concentric with the ellipse

x 2 y2   1 . Prove that the a 2 b2

jor axis at an angle

22.

 r 2  b2 tan 1  2 2  a r 

   

18.

23.

 +  + r + s = 2n. 24.

A point P moves so that a circle with PA as diameter where A  (a, 0) touches the circle x2 + y2 = 4a2 internally. Show that locus of P is an ellipse with eccentricity ½.

25.

The tangent at a point P (a cos , b sin ) of an ellipse

ellipse & G is point of intersection of normal with x-axis. 19.

If , , r & s be the eccentric angles of four concyclic

x 2 y2 points on the ellipse 2  2  1 , then prove that a b

The normal GP is produced to Q, so that GQ = n . GP. Prove that the locus of Q is the ellipse

x2 y2   1 . P is a point of the a 2 (n 2  e 2  ne 2 ) n 2 b 2

Find the locus of the middle points of chords of an ellipse (a) the tangents at the ends of which intersect at right angles. (b) which subtends a right angle at the centre.

and find its length. The tangent at P meets the axis in T & t and CY is the perpendicular on it from the centre; prove that (a) Tt . PY = a2 – b2 and (b) the least value of Tt is a + b.

Find the tangent of the angle between CP and the normal at P and prove that its greatest value is

a 2  b2 . 2ab

common tangents is inclined to the ma

17.

Prove that, in general four normals can be drawn

x 2 y2 to the ellipse 2  2  1 and the a b

intersection is (b)

20.

x 2 y2   1 , meets its auxiliary circle a 2 b2

in two points, the chord joining which subtends a right angle the centre. Show that the eccentricity of the ellipse is (1 + sin2)–½.

If P be a point on the ellipse, whose ordinates is y’, prove that the angle between the tangent at P and 2  1  b  tan the focal distance of P is  aey  .  

ANSWERS SUBJECTIVE (INITIAL STEP EXERCISE)

2.

 x 2 y2   1 1  x 2 y 2   4  4    2  2  2  2  b  a b  a b  a

5.

5x2 + 9y2 – 54y + 36 = 0

10.

2 2

2

2

2

al +bm =n

2

2 1 ;e  ) 3 3

3.

(1,2) ; 2 ,

6.

(x2 + y2)2 = a2x2 + b2y2

20.

x 2 y2  2  r2 2 a b

ANSWERS SUBJECTIVE (FINAL STEP EXERCISE)

1.

 a 2 4b 2   2  2   9 y  x

Einstein Classes,

2.

x 2 y2  2 a 2 b2


Ellipse

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