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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Find the equation of chord joining two points of hyperbola which passes through parametric coordinates, $(3 \sec θ, 2 \tan θ) and (3 \sec ϕ, 2 \tan ϕ)$ .

Important Questions on Hyperbola

MEDIUM

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

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:

EASY

Mathematics>Coordinate Geometry>Hyperbola>Basics of Hyperbola

If the eccentricity of a hyperbola is

\frac{5}{3},

then the eccentricity of its conjugate hyperbola is

MEDIUM

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

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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:

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Basics of 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>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:

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Basics of Hyperbola

The equation of the directrices of the hyperbola

3 x^{2} - 3 y^{2} - 18 x + 12 y + 2 = 0

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Basics of 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

EASY

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

The distance between the foci of a hyperbola is

16

and its eccentricity is

\sqrt{2}

. Its equation can be

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

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Basics of 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>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:

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:

Find the equation of chord joining two points of hyperbola which passes through parametric coordinates, (3secθ,2tanθ) and (3secϕ,2tanϕ).

Important Questions on Hyperbola

Learn and Prepare for any exam you want !

Find the equation of chord joining two points of hyperbola which passes through parametric coordinates, $(3 \sec θ, 2 \tan θ) and (3 \sec ϕ, 2 \tan ϕ)$ .