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

If line x + y = 1 is intersecting ellipse x 2 3 + y 2 4 = 1 at two distinct points A and B , then point of intersection of tangents at A and B :

Lies on circle x 2 + y 2 - 6 x - 6 y + 17 = 0 .

HARD

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The area of the parallelogram formed by the tangents at the ends of conjugate diameters of an ellipse is

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Important Questions on Ellipse

MEDIUM

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

Let the tangents at the points

P

and

Q

on the ellipse

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

meet at the point

R (\sqrt{2}, 2 \sqrt{2} - 2)

. If

S

is the focus of the ellipse on its negative major axis, then

S P^{2} + S Q^{2}

is equal to

HARD

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

Let

E

be the ellipse

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

. For any three distinct points

P, Q

and

Q^{'}

E

, let

M (P, Q)

be the mid-point of the line segment joining

P

and

Q

, and

M (P, Q^{'})

be the mid-point of the line segment joining

P

and

Q^{'}

. Then the maximum possible value of the distance between

M (P, Q)

and

M (P, Q^{'})

, as

P, Q

and

Q^{'}

vary on

E

, 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

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

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

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

Prove that the locus of the intersection of normals at the ends of conjugate diameters of an ellipse

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

is the curve

2 {(a^{2} x^{2} + b^{2} y^{2})}^{3} = {(a^{2} - b^{2})}^{2} {(a^{2} x^{2} - b^{2} y^{2})}^{2} .

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

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

If the chords of constant of tangents from two points

(x_{1}, y_{1})

and

(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

MEDIUM

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

2 y = x

and

3 y + 4 x = 0

are the equations of a pair of conjugate diameters of an ellipse, then 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 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

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

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

In an ellipse

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

, show that the perpendiculars from the center upon all the chords which join the ends of the perpendicular diameters, are of constant length.

HARD

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

Tangents are drawn from the points on the line

x - y - 5 = 0

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

, then all the chords of contact pass through a fixed point, whose co-ordinates are -

MEDIUM

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)

is:

HARD

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

If line $x + y = 1$ is intersecting ellipse $\frac{x^{2}}{3} + \frac{y^{2}}{4} = 1$ at two distinct points $A$ and $B$ , then point of intersection of tangents at $A$ and $B$ :

HARD

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

If a variable tangent of the circle

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

intersects the ellipse

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

at points

P

and

Q

, then the locus of the point of intersection of tangent at

P

and

Q

EASY

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

Cotyledons are also called-

The area of the parallelogram formed by the tangents at the ends of conjugate diameters of an ellipse is

Important Questions on Ellipse

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