Alpha Question Bank for Engineering: Mathematics>Chapter 19 - Hyperbola>Exercise-2>Q 1

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If $e$ and $e^{'}$ are the eccentricities of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ and $\frac{y^{2}}{b^{2}} - \frac{x^{2}}{a^{2}} = 1$ , then the point $(\frac{1}{e}, \frac{1}{e^{'}})$ lies on the circle

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Important Questions on Hyperbola

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Alpha Question Bank for Engineering: Mathematics>Chapter 19 - Hyperbola>Exercise-2>Q 2

P

is a point on the hyperbola

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

is the foot of the perpendicular from

P

on the transverse axis. The tangent to the hyperbola at

P

meets the transverse axis at

T .

If

O

is the centre of the hyperbola, then

O T

.

O N

is equal to

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Alpha Question Bank for Engineering: Mathematics>Chapter 19 - Hyperbola>Exercise-2>Q 3

Tangent at any point on the hyperbola

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

cut the axes at

A

and

B

, respectively. If the rectangle

O A P B

(where

O

is origin) is completed then locus of point

P

is given by

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Alpha Question Bank for Engineering: Mathematics>Chapter 19 - Hyperbola>Exercise-2>Q 4

If the chord of contact of tangents from two points

(x_{1}, y_{1})

and

(x_{2}, y_{2})

to the hyperbola

\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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Alpha Question Bank for Engineering: Mathematics>Chapter 19 - Hyperbola>Exercise-2>Q 5

The sides

A C

and

A B

of a triangle

A B C

touch the conjugate hyperbola of the hyperbola

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

at

C

and

B

respectively. If the vertex

A

lies on the ellipse

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

, the side

B C

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Alpha Question Bank for Engineering: Mathematics>Chapter 19 - Hyperbola>Exercise-2>Q 6

Tangents are drawn from any point on the hyperbola

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

to the circle

x^{2} + y^{2} = 9,

then the locus of midpoint of the chord of contact is

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Alpha Question Bank for Engineering: Mathematics>Chapter 19 - Hyperbola>Exercise-2>Q 7

If

A B

is a double ordinate of the hyperbola

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

such that

O A B

(

O

is the origin) is an equilateral triangle, then the eccentricity

' e'

of the hyperbola satisfies:

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Alpha Question Bank for Engineering: Mathematics>Chapter 19 - Hyperbola>Exercise-2>Q 14

The length of that focal chord of the hyperbola

x y = 8

which touches the circle

x^{2} + y^{2} = 8

is.

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Alpha Question Bank for Engineering: Mathematics>Chapter 19 - Hyperbola>Exercise-2>Q 15

The sum of lengths of perpendiculars drawn from focii to any real tangent to the hyperbola

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

is always greater than

a

, then find maximum value of

a

.