Complete Study Pack for Engineering Entrances Objective Mathematics Vol 1>Chapter 15 - Ellipse>Work Book Exercise 15.2>Q 5

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The slope of a common tangent to the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ and a concentric circle of radius $r$ is

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Important Questions on Ellipse

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IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 1>Chapter 15 - Ellipse>Work Book Exercise 15.2>Q 6

If

p

and

p^{'}

denote the lengths of the perpendicular from a focus and the centre of an ellipse with semi-major axis of length

a

, respectively on a tangent to the ellipse and

r

denotes the focal distance of the point, then

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Complete Study Pack for Engineering Entrances Objective Mathematics Vol 1>Chapter 15 - Ellipse>Work Book Exercise 15.2>Q 7

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

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Complete Study Pack for Engineering Entrances Objective Mathematics Vol 1>Chapter 15 - Ellipse>Work Book Exercise 15.3>Q 1

Chords of an ellipse are drawn through the positive end of the minor axis. Then, their midpoint lies on

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Complete Study Pack for Engineering Entrances Objective Mathematics Vol 1>Chapter 15 - Ellipse>Work Book Exercise 15.3>Q 2

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

π / 4

is

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Complete Study Pack for Engineering Entrances Objective Mathematics Vol 1>Chapter 15 - Ellipse>Work Book Exercise 15.3>Q 3

If the chords of contact of tangent 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

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Complete Study Pack for Engineering Entrances Objective Mathematics Vol 1>Chapter 15 - Ellipse>Work Book Exercise 15.3>Q 4

The locus of the point of intersection of perpendicular tangents to

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

and

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

is

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Complete Study Pack for Engineering Entrances Objective Mathematics Vol 1>Chapter 15 - Ellipse>Work Book Exercise 15.3>Q 5

The locus of the midpoints of the chords $2 x + 3 y + λ = 0$ of the ellipse

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

,

λ

being a parameter, is

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Complete Study Pack for Engineering Entrances Objective Mathematics Vol 1>Chapter 15 - Ellipse>Work Book Exercise 15.3>Q 6

In the ellipse

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

the length of perpendiculars from the centre upon all chords which join ends of perpendicular diameters, is