Let P is a point of the ellipse x 2 a 2 + y 2 b 2 = 1 , whose foci are S and S ' . If ∠ P S S ' = α and ∠ P S ' S = β , then

tan α 2 tan β 2 = a 2 - b 2 b 2 a - a 2 - b 2 when a > b

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The tangent at any point $P$ on a standard ellipse with foci as $S$ and $S^{'}$ meets the tangents at the vertices $A$ and $A'$ in the points $V$ and $V'$ , then -

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Important Questions on Ellipse

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IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 19 - Ellipse>EXERCISE-2>Q 4

The area of the rectangle formed by the perpendiculars from the centre of the standard ellipse to the tangent and normal at its point whose eccentric angle is

\frac{π}{4}

is -

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Beta Question Bank for Engineering: Mathematics>Chapter 19 - Ellipse>EXERCISE-2>Q 5

Q

is a point on the auxiliary circle of an ellipse. P is the corresponding point on ellipse.

N

is the foot of perpendicular from focus

S

, to the tangent of auxiliary circle at

Q

, then -

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Beta Question Bank for Engineering: Mathematics>Chapter 19 - Ellipse>EXERCISE-2>Q 6

The line,

l x + m y + n = 0

will cut the ellipse

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

in points whose eccentric angles differ by

\frac{π}{2}

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Beta Question Bank for Engineering: Mathematics>Chapter 19 - Ellipse>EXERCISE-2>Q 7

A circle has the same centre as an ellipse and passes through the foci

F_{1}

and

F_{2}

of the ellipse, such that the two curves intersect at four points. Let

P

be any one of their point of intersection. If the major axis of the ellipse is

17

units and the area of triangle

P F_{1} F_{2}

30

sq. units, then the distance between the foci is

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Beta Question Bank for Engineering: Mathematics>Chapter 19 - Ellipse>EXERCISE-2>Q 8

The normal at a variable point

P

on the ellipse

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

of eccentricity

e

meets the axes of the ellipse at

Q

and

R,

then the locus of the midpoint of

Q R

is a conic with an eccentricity

e^{'}

such that

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Beta Question Bank for Engineering: Mathematics>Chapter 19 - Ellipse>EXERCISE-2>Q 9

The length of the normal (terminated by the major axis) at a point of the ellipse

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

is -

where $r$ and $r_{1}$ are the focal distance of the point.

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Beta Question Bank for Engineering: Mathematics>Chapter 19 - Ellipse>EXERCISE-2>Q 10

Point

' O'

is the centre of the ellipse with major axis

A B

and minor axis

C D

. Point

F

is one focus of the ellipse. If

O F = 6

and the diameter of the inscribed circle of triangle

O C F

2,

then the product

(A B) (C D)

is equal to -

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IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 19 - Ellipse>EXERCISE-2>Q 11

Let

P

is a point of the ellipse

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

whose foci are

S

and

S^{'} .

∠ P S S^{'} = α

and

∠ P S^{'} S = β,

then

The tangent at any point P on a standard ellipse with foci as S and S' meets the tangents at the vertices A and A' in the points V and V', then -

Important Questions on Ellipse

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The tangent at any point $P$ on a standard ellipse with foci as $S$ and $S^{'}$ meets the tangents at the vertices $A$ and $A'$ in the points $V$ and $V'$ , then -