Alpha Question Bank for Engineering: Mathematics>Chapter 19 - Hyperbola>Exercise-1>Q 41

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The locus of the midpoints of the chords passing through a fixed point $(α, β)$ of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ is

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Important Questions on Hyperbola

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Alpha Question Bank for Engineering: Mathematics>Chapter 19 - Hyperbola>Exercise-1>Q 42

If the normal at

(c t, \frac{c}{t})

on the curve

x y = c^{2}

meets the curve again at

t^{'}

, then

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Alpha Question Bank for Engineering: Mathematics>Chapter 19 - Hyperbola>Exercise-1>Q 43

The locus of the foot of perpendicular drawn from the centre of the hyperbola

x^{2} - y^{2} = 25

to its normal.

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Alpha Question Bank for Engineering: Mathematics>Chapter 19 - Hyperbola>Exercise-1>Q 44

Equation of common tangent to ellipse

5 x^{2} + 2 y^{2} = 10

, and hyperbola

11 x^{2} - 3 y^{2} = 33

is

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

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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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