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KVPY Aptitude Test - Stream SA

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Let $d_{1}$ and $d_{2}$ be the lengths of the perpendiculars drawn from foci $S$ and $S^{'}$ of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ to the tangent at any point $P$ on the ellipse. Then, $S P : S^{'} P =$

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Important Questions on Conic Section

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Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 12 - Conic Section>Exercise>Q 38

There are exactly two points on the ellipse

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

whose distance from its centre is the same and is equal to

\frac{\sqrt{a^{2} + 2 b^{2}}}{2} .

Then the eccentricity of the ellipse is

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Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 12 - Conic Section>Exercise>Q 39

The eccentricity of locus of point

(3 h + 2, k)

, where

(h, k)

lies on the circle

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

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Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 12 - Conic Section>Exercise>Q 40

Let

S

and

S^{'}

be two foci of the ellipse

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

. If a circle described on

S S^{'}

as diameter intersects the ellipse in real and distinct points, then the eccentricity

e

of the ellipse satisfies

HARD

KVPY Aptitude Test - Stream SA

IMPORTANT

Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 12 - Conic Section>Exercise>Q 41

From any point $P$ lying in first quadrant on the ellipse $\frac{x^{2}}{25} + \frac{y^{2}}{16} = 1, P N$ is drawn perpendicular to the major axis and produced at $Q$ so that $N Q$ equals to $P S,$ where $S$ is a focus. Then the locus of $Q$ is

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Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 12 - Conic Section>Exercise>Q 42

The slopes of the common tangents of the ellipse

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

and the circle

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

are

HARD

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IMPORTANT

Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 12 - Conic Section>Exercise>Q 43

Tangents are drawn to the ellipse

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

and the circle

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

at the points where a common ordinate cuts them (on the same side of the

x

-axis). Then the greatest acute angle between these tangents is given by

HARD

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Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 12 - Conic Section>Exercise>Q 44

The point of intersection of the tangents at the point

P

on the ellipse

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

and its corresponding point

Q

on the auxiliary circle meet on the line

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Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 12 - Conic Section>Exercise>Q 45

If the tangents to the ellipse

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

make angles

α

and

β

with the major axis such that

\tan α + \tan β =

λ,

then the locus of their point of intersection is

Let d1 and d2 be the lengths of the perpendiculars drawn from foci S and S' of the ellipse x2a2+y2b2=1 to the tangent at any point P on the ellipse. Then, SP:S'P =

Important Questions on Conic Section

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Let $d_{1}$ and $d_{2}$ be the lengths of the perpendiculars drawn from foci $S$ and $S^{'}$ of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ to the tangent at any point $P$ on the ellipse. Then, $S P : S^{'} P =$