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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Let the eccentricity of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ be the reciprocal to the eccentricity of the ellipse $x^{2} + 4 y^{2} = 4$ . If the hyperbola passes through a focus of the ellipse, then find the value of $100 a .$ $(\sqrt{3} = 1.73)$

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Important Questions on 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:

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

A hyperbola with centre at

(0, 0)

has its transverse axis along

X

-axis whose length is

12

. If

(8, 2)

is a point on the hyperbola, then its eccentricity is

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

For the hyperbola

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

which of the following remains fixed when

α

varies?

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 ?

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:

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

A double ordinate

P Q

of the hyperbola

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

is such that

Δ O P Q

is equilateral

O

being the centre of the hyperbola. Then the eccentricity

e

satisfies the relation

EASY

Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

Let the eccentricity of the hyperbola

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

be reciprocal to that of the ellipse

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

then the ratio

a^{2} : b^{2}

equals

HARD

Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

What will be the equation of the standard hyperbola where foci are

(0, \pm 10)

and the length of the latus rectum is

30 ?

MEDIUM

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

MEDIUM

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

The locus of the point of intersection of the lines

(\sqrt{3}) k x + k y - 4 \sqrt{3} = 0

and

\sqrt{3} x - y - 4 (\sqrt{3}) k = 0

is a conic, whose eccentricity is

MEDIUM

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

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

The locus of the centroid of the triangle formed by any point

P

on the hyperbola

16 x^{2} - 9 y^{2} + 32 x + 36 y - 164 = 0

and its foci is

EASY

Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

The length of conjugate axis of a hyperbola is greater than the length of transverse axis. Then, the eccentricity

e

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

Let

θ \in (0, \frac{π}{2}) .

If the eccentricity of the hyperbola

x^{2} \cos^{2} θ - y^{2} = 6 \cos^{2} θ

\sqrt{3}

times the eccentricity of the ellipse

x^{2} + y^{2} \cos^{2} θ = 30 \cos^{2} θ,

then

θ

is equal to

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

If one focus of a hyperbola is

(3, 0)

, the equation of its directrix is

4 x - 3 y - 3 = 0

and its eccentricity

e = \frac{5}{4}

, then the co-ordinates of its vertex is

EASY

Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

The eccentricity of the hyperbola

\frac{x^{2}}{49} - \frac{y^{2}}{576} = 1

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

If the eccentricity of a conic satisfies the equation

2 x^{3} + 10 x - 13 = 0

, then that conic is

MEDIUM

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

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

?

Let the eccentricity of the hyperbola x2a2-y2b2=1 be the reciprocal to the eccentricity of the ellipse x2+4y2=4. If the hyperbola passes through a focus of the ellipse, then find the value of 100a. (3=1.73)

Important Questions on Hyperbola

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Let the eccentricity of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ be the reciprocal to the eccentricity of the ellipse $x^{2} + 4 y^{2} = 4$ . If the hyperbola passes through a focus of the ellipse, then find the value of $100 a .$ $(\sqrt{3} = 1.73)$