The chords of contact of perpendicular tangents to the ellipse x 2 a 2 + y 2 b 2 = 1 touch another fixed ellipse whose equation is

x 2 a 4 + y 2 b 4 = 1 a 2 + b 2

x 2 a 2 + y 2 b 2 = 1 2 a 2 + b 2

x 2 a 2 + y 2 b 2 = 2 ( a 2 - b 2 )

x 2 a 2 - y 2 b 2 = 2 3 a 2 - b 2

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Find the equation of the diameter of an ellipse $2 x^{2} + 3 y^{2} = 6$ conjugate to the diameter $3 y + 2 x = 0$

Important Questions on Ellipse

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Mathematics>Coordinate Geometry>Ellipse>Pair of Tangents, Chord of Contact and Chord with Midpoint of an Ellipse

S

and

S^{'}

are the focii of an ellipse,

B

is one end of the minor axis and

∠ S B S' = 90 °

, then the eccentricity of that ellipse is

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Pair of Tangents, Chord of Contact and Chord with Midpoint of an Ellipse

Let

x^{2} = 4 k y, k > 0

be a parabola with vertex

A

. Let

B C

be its latusrectum. An ellipse with centre on

B C

touches the parabola at

A

, and cuts

B C

at points

D

and

E

such that

B D = D E = E C

(

B, D, E, C

in that order). The eccentricity of the ellipse is

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Pair of Tangents, Chord of Contact and Chord with Midpoint of an Ellipse

Let

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (b < a),

be an ellipse with major axis

A B

and minor axis

C D .

Let

F_{1}

and

F_{2}

be its two foci, with

A, F_{1}, F_{2}, B

in that order on the segment

A B .

Suppose

∠ F_{1} C B = 90 ° .

The eccentricity of the ellipse is

HARD

Mathematics>Coordinate Geometry>Ellipse>Pair of Tangents, Chord of Contact and Chord with Midpoint of an Ellipse

The eccentricity of an ellipse having centre at the origin, axes along the co-ordinate axes and passing through the points

(4, - 1)

and

(- 2, 2)

HARD

Mathematics>Coordinate Geometry>Ellipse>Pair of Tangents, Chord of Contact and Chord with Midpoint of an Ellipse

e_{1}

and

e_{2}

are the eccentricities of the ellipse

\frac{x^{2}}{18} + \frac{y^{2}}{4} = 1

and the hyperbola

\frac{x^{2}}{9} - \frac{y^{2}}{4} = 1

respectively and

(e_{1}, e_{2})

is a point on the ellipse

15 x^{2} + 3 y^{2} = k

, then the value of

k

is equal to

HARD

Mathematics>Coordinate Geometry>Ellipse>Pair of Tangents, Chord of Contact and Chord with Midpoint of an Ellipse

Let

a, b

and

λ

be positive real numbers. Suppose

P

is an end point of the latus rectum of the parabola

y^{2} = 4 λ x

, and suppose the ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

passes through the point

P

. If the tangents to the parabola and the ellipse at the point

P

are perpendicular to each other, then the eccentricity of the ellipse is

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Pair of Tangents, Chord of Contact and Chord with Midpoint of an Ellipse

Let

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, a > b

be an ellipse with foci

F_{1}

and

F_{2}

Let

A O

be its semi-minor axis, where

O

is the centre of the ellipse. The lines

A F_{1}

and

A F_{2}

, when extended, cut the ellipse again at points

B

and

C

respectively. Suppose that the

Δ A B C

is equilateral. Then, the eccentricity of the ellipse is

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Mathematics>Coordinate Geometry>Ellipse>Pair of Tangents, Chord of Contact and Chord with Midpoint of an Ellipse

For some

θ \in (0, \frac{π}{2}),

if the eccentricity of the hyperbola,

x^{2} - y^{2} \sec^{2} θ = 10

\sqrt{5}

times the eccentricity of the ellipse,

x^{2} \sec^{2} θ + y^{2} = 5,

then the length of the latus rectum of the ellipse, is

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Mathematics>Coordinate Geometry>Ellipse>Pair of Tangents, Chord of Contact and Chord with Midpoint of an Ellipse

O B

is the semi-minor axis of an ellipse,

F_{1}

and

F_{2}

are its focii and the angle between

F_{1} B

and

F_{2} B

is a right angle, then the square of the eccentricity of the ellipse is

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Pair of Tangents, Chord of Contact and Chord with Midpoint of an Ellipse

The equation of a diameter conjugate to a diameter

y = \frac{b}{a} x

of the ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

, is

HARD

Mathematics>Coordinate Geometry>Ellipse>Pair of Tangents, Chord of Contact and Chord with Midpoint of an Ellipse

The chords of contact of perpendicular tangents to the ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

touch another fixed ellipse whose equation is

HARD

Mathematics>Coordinate Geometry>Ellipse>Pair of Tangents, Chord of Contact and Chord with Midpoint of an Ellipse

From a point on the line

(t + 2) (x + y) = 1, t \neq - 2

, tangents are drawn to the ellipse

4 x^{2} + 16 y^{2} = 1

. It is given that the chord of contact passes through a fixed point. Then the number of integral values of '

t

' for which the fixed point always lies inside the ellipse is

HARD

Mathematics>Coordinate Geometry>Ellipse>Pair of Tangents, Chord of Contact and Chord with Midpoint of an Ellipse

The chord of contact of the tangents drawn from

(α, β)

to an ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

touches the circle

x^{2} + y^{2} = c^{2}

, then the locus of

(α, β)

HARD

Mathematics>Coordinate Geometry>Ellipse>Pair of Tangents, Chord of Contact and Chord with Midpoint of an Ellipse

If the point of intersection of the ellipses

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

and

\frac{x^{2}}{α^{2}} + \frac{y^{2}}{β^{2}} = 1

be at the extremities of the conjugate diameters of the former, then -

HARD

Mathematics>Coordinate Geometry>Ellipse>Pair of Tangents, Chord of Contact and Chord with Midpoint of an Ellipse

The maximum distance of the centre of the ellipse $\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$ from the chord of contact of mutually perpendicular tangents of the ellipse is

HARD

Mathematics>Coordinate Geometry>Ellipse>Pair of Tangents, Chord of Contact and Chord with Midpoint of an Ellipse

If the chords of contact of tangents from two points

(x_{1}, y_{1})

(x_{2}, y_{2})

to the ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

are at right angles then

(\frac{x_{1} x_{2}}{y_{1} y_{2}})

is equal to

HARD

Mathematics>Coordinate Geometry>Ellipse>Pair of Tangents, Chord of Contact and Chord with Midpoint of an Ellipse

From a point

P

perpendicular tangents

PQ

and

PR

are drawn to ellipse

x^{2} + 4 y^{2} = 4

, then locus of circumcentre of the triangle

PQR

HARD

Mathematics>Coordinate Geometry>Ellipse>Pair of Tangents, Chord of Contact and Chord with Midpoint of an Ellipse

Which of the following options is most revalent?

Statement 1:
Let $P$ be any point on a directrix of an ellipse. Then, the chords of contact of the point $P$ with respect to the ellipse and its auxiliary circle intersect at the corresponding focus.

Statement 2:
The equation of the family of lines passing through the point of intersection of lines $L_{1} = 0$ and $L_{2} = 0$ is $L_{1} + λ L_{2} = 0$ .

HARD

Mathematics>Coordinate Geometry>Ellipse>Pair of Tangents, Chord of Contact and Chord with Midpoint of an Ellipse

Find the inclination to the major axis of the diameter of the ellipse the square of whose length is the arithmetical mean between the squares on the major and minor axes.

HARD

Mathematics>Coordinate Geometry>Ellipse>Pair of Tangents, Chord of Contact and Chord with Midpoint of an Ellipse

The length of the diameter of the ellipse

\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1

perpendicular to the asymptotes of the hyperbola

\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1

passing through the first and third quadrant is

Find the equation of the diameter of an ellipse 2x2+3y2=6 conjugate to the diameter 3y+2x=0

Important Questions on Ellipse

Learn and Prepare for any exam you want !

Find the equation of the diameter of an ellipse $2 x^{2} + 3 y^{2} = 6$ conjugate to the diameter $3 y + 2 x = 0$