Let S = x , y ∈ R 2 : y 2 1 + r - x 2 1 - r = 1 , where r ≠ ± 1 . Then S represents:

An ellipse whose eccentricity is 2 r + 1 , when r > 1

An ellipse whose eccentricity is 1 r + 1 , when r > 1 .

A hyperbola whose eccentricity is 2 r + 1 , when 0 < r < 1 .

A hyperbola whose eccentricity is 2 1 - r , when 0 < r < 1

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The length of the latus-rectum of a hyperbola is half the length of its transverse axis. The eccentricity of the hyperbola is

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Important Questions on Hyperbola

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

The locus of the point of intersection of the straight lines,

t x - 2 y - 3 t = 0

and

x - 2 t y + 3 = 0 (t \in R),

is:

HARD

Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

Equation

\frac{1}{r} = \frac{1}{8} + \frac{3}{8} \cos θ

represents

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

Which of the following is the equation of a hyperbola?

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Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

Let

a

and

b

respectively be the semi-transverse and semi-conjugate axes of a standard hyperbola whose eccentricity satisfies the equation

9 e^{2} - 18 e + 5 = 0

. If

S (5, 0)

is a focus and

5 x = 9

is the corresponding directrix of this hyperbola, then

a^{2} - b^{2}

is equal to

EASY

Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

If the eccentricity of the hyperbola

x^{2} - y^{2} {cosec}^{2} α = 25

\sqrt{5}

times the eccentricity of the ellipse

x^{2} {cosec}^{2} α + y^{2} = 5,

then

α

is equal to

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

A hyperbola whose transverse axis is along the major axis of the conic

\frac{x^{2}}{3} + \frac{y^{2}}{4} = 4

and has vertices at the foci of the conic. If the eccentricity of the hyperbola is

\frac{3}{2}

, then which of the following points does not lie on the hyperbola

?

HARD

Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

The foci of the hyperbola

16 x^{2} - 9 y^{2} - 64 x + 18 y - 90 = 0

are

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Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

The eccentricity of the hyperbola whose length of the latus rectum is equal to

8

and the length of its conjugate axis is equal to half of the distance between its foci, is

EASY

Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

If the vertices of a hyperbola be at

(- 2, 0)

and

(2, 0)

and one of its foci be at

(- 3, 0)

, then which one of the following points does not lie on this hyperbola ?

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

Let

S = \{(x, y) \in R^{2} : \frac{y^{2}}{1 + r} - \frac{x^{2}}{1 - r} = 1\},

where

r \neq \pm 1 .

Then

S

represents:

EASY

Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

5 x + 9 = 0

is the directrix of the hyperbola

16 x^{2} - 9 y^{2} = 144,

then its corresponding focus is:

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Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

Let

0 < θ < \frac{π}{2} .

If the eccentricity of the hyperbola

\frac{x^{2}}{\cos^{2} θ} - \frac{y^{2}}{\sin^{2} θ} = 1

is greater than

2,

then the length of its latus rectum lies in the interval:

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

If a directrix of a hyperbola centered at the origin and passing through the point

(4, - 2 \sqrt{3})

5 x = 4 \sqrt{5}

and its eccentricity is

e

, then:

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Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

The eccentricity of the hyperbola whose length of its conjugate axis is equal to half of the distance between its foci, is

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Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

The latus rectum of the hyperbola

3 x^{2} - 2 y^{2} = 6

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Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

If equation

(10 x - 5)^{2} + (10 y - 4)^{2} = λ^{2} (3 x + 4 y - 1)^{2}

represents a hyperbola, then

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

e_{1}

and

e_{2}

are the eccentricities of a hyperbola

3 x^{2} - 3 y^{2} = 25

and its conjugate, then

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Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

The vertices of the hyperbola are at

(- 5, - 3)

and

(- 5, - 1)

and the extremities of the conjugate axis are at

(- 7, - 2)

and

(- 3, - 2),

then the equation of the hyperbola is

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Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

A hyperbola has its centre at the origin, passes through the point

(4, 2)

and has transverse axis of length

4

along the

x - a x i s .

Then the eccentricity of the hyperbola is:

EASY

Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

The equation of the hyperbola with vertices

(3, 0), (- 3, 0)

and semi-latus rectum

4

is given by:

The length of the latus-rectum of a hyperbola is half the length of its transverse axis. The eccentricity of the hyperbola is

Important Questions on Hyperbola

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