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

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The equation $y^{2} - x^{2} = 4$ represents a rectangular hyperbola.

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Important Questions on Hyperbola

EASY

Mathematics>Coordinate Geometry>Hyperbola>Basics of 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>Basics of Hyperbola

Let

R

be a rectangle given by the lines

x = 0, x = 2

y = 0

and

y = 5

. Let

A (α, 0)

and

B (0, β), α \in [0, 2]

and

β \in [0, 5]

, be such that the line segment

A B

divides the area of the rectangle

R

in the ratio

4 : 1

. Then, the mid-point of

A B

lies on a

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Basics of 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>Basics of 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>Basics of Hyperbola

The coordinate of the vertices of the rectangular hyperbola $x y = 16$ .

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Basics of Hyperbola

Which of the following is the equation of a hyperbola?

EASY

Mathematics>Coordinate Geometry>Hyperbola>Basics of 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>Basics of Hyperbola

If the eccentricity of the standard hyperbola passing through the point

(4,6)

2,

then the equation of the tangent to the hyperbola at

(4,6)

is:

HARD

Mathematics>Coordinate Geometry>Hyperbola>Basics of Hyperbola

Find the equation of the rectangular hyperbola which passes through the points $(6, 0)$ and $(- 3, 0)$ and has an asymptote $x + 2 y - 5 = 0$ .

HARD

Mathematics>Coordinate Geometry>Hyperbola>Basics of Hyperbola

The length of the latus rectum of the rectangular hyperbola

x y = 32

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Basics of 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>Basics of Hyperbola

A square

A B C D

has all its vertices on the curve

x^{2} y^{2} = 1 .

The midpoints of its sides also lie on the same curve. Then, the square of area of

A B C D

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Basics of Hyperbola

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

HARD

Mathematics>Coordinate Geometry>Hyperbola>Basics of Hyperbola

The foci of the hyperbola

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

are

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Basics of 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>Basics of 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:

EASY

Mathematics>Coordinate Geometry>Hyperbola>Basics of 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>Basics of Hyperbola

Let the eccentricity of an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ is reciprocal to that of the hyperbola $2 x^{2} - 2 y^{2} = 1$ . If the ellipse intersects the hyperbola at right angles, then square of length of the latus-rectum of the ellipse is _____.

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Basics of Hyperbola

Show that the tangent to a rectangular hyperbola terminated by its asymptotes is bisected at the point of contact.

HARD

Mathematics>Coordinate Geometry>Hyperbola>Basics of Hyperbola

On a rectangular hyperbola

x^{2} - y^{2} = a^{2}, a > 0,

three points

A, B, C

are taken as follows

: A = (- a, 0); B

and

C

are placed symmetrically with respect to the

X

-axis on the branch of the hyperbola not containing

A

. Suppose that the

Δ A B C

is equilateral. If the side length of the

Δ A B C

k a,

then

k

lies in the interval

The equation y2-x2=4 represents a rectangular hyperbola.

Important Questions on Hyperbola

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The equation $y^{2} - x^{2} = 4$ represents a rectangular hyperbola.