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

MEDIUM

Earn 100

Prove that the locus of the mid-points of the chords of a hyperbola passing through a fixed point ( $h, k$ ) is again a hyperbola whose centre is at the point $(\frac{h}{2}, \frac{k}{2})$ .

Important Questions on 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 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

?

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

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

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

EASY

Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

The latus rectum of the hyperbola

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

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

If the eccentricity of a conic satisfies the equation

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

, then that conic is

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

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

Which of the following is the equation of a hyperbola?

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

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

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

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:

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

The value of

b^{2}

in order that the foci of the hyperbola

\frac{x^{2}}{144} - \frac{y^{2}}{81} = \frac{1}{25}

and the ellipse

\frac{x^{2}}{16} + \frac{y^{2}}{b^{2}} = 1

coincide is

EASY

Mathematics>Coordinate Geometry>Hyperbola>Equation of a Hyperbola

If the eccentricity of a hyperbola is

\frac{5}{3},

then the eccentricity of its conjugate hyperbola is

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

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:

EASY

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

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 ?

Prove that the locus of the mid-points of the chords of a hyperbola passing through a fixed point (h, k) is again a hyperbola whose centre is at the point h2, k2.

Important Questions on Hyperbola

Learn and Prepare for any exam you want !

Prove that the locus of the mid-points of the chords of a hyperbola passing through a fixed point ( $h, k$ ) is again a hyperbola whose centre is at the point $(\frac{h}{2}, \frac{k}{2})$ .