If a hyperbola whose foci are S ≡ 2 , 4 and S ' ≡ 8 , - 2 touches x -axis, then equation of hyperbola is

x - y - 4 2 20 - x + y - 6 2 16 = 1

x + y - 6 2 10 - x - y - 4 2 8 = 1

x - y - 4 2 10 - x + y - 6 2 8 = 1

x + y - 6 2 20 - x - y - 4 2 16 = 1

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Tangent at any point $P$ on the hyperbola $\frac{x^{2}}{9} - \frac{y^{2}}{4} = 1$ intersects the asymptotes at points $A$ and $B,$ if $C$ is the centre of the hyperbola, then area of $Δ ABC$ is

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Important Questions on Hyperbola

HARD

Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

A hyperbola having the transverse axis of length,

\sqrt{2}

has the same foci as that of the ellipse,

3 x^{2} + 4 y^{2} = 12

then this hyperbola does not pass through which of the following points?

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

If the foci of the ellipse

16 x^{2} + 7 y^{2} = 112

and of the hyperbola

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

coincide then

a =

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

If the sum of slopes of concurrent normals to the curve

x y = 4

is equal to the sum of ordinates of conormal points then locus of

P

HARD

Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

If a hyperbola whose foci are

S \equiv (2, 4)

and

S^{'} \equiv (8, - 2)

touches

x

-axis, then equation of hyperbola is

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

If ellipse

\frac{x^{2}}{49} + \frac{y^{2}}{36} = 1

and hyperbola

\frac{x^{2}}{9 + λ^{2}} - \frac{y^{2}}{3 + λ} = 1

are confocal then the possible of

λ

is -

EASY

Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

The ellipse

4 x^{2} + 9 y^{2} = 36

and the hyperbola

4 x^{2} - y^{2} = 4

have the same foci, and they intersect at right then the equation of the circle through the point of intersection of two conics is -

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b), x^{2} - y^{2} = c^{2}

are orthogonal then

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

The product of the lengths of the perpendiculars drawn from foci on any tangent to the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ is

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

The product of perpendicular drawn from any point on

\frac{x^{2}}{9} - \frac{y^{2}}{16} = 1

upon its asymptote is-

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

Tangent at any point

P

on the hyperbola

\frac{x^{2}}{9} - \frac{y^{2}}{4} = 1

intersects the asymptotes at points

A

and

B

, if

C

is the centre of the hyperbola, then area of

Δ A B C

HARD

Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

An ellipse

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

and the hyperbola

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

intersect orthogonally. It is given that the eccentricity of the ellipse is reciprocal of that of hyperbola, then

\frac{a^{2}}{b^{2}}

is equal to

HARD

Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

If a ray of light incident along the line

\sqrt{231} x + 5 y = 6 \sqrt{231}

gets reflected from the hyperbola

\frac{x^{2}}{25} - \frac{y^{2}}{11} = 1

, then its reflected ray goes along the line

HARD

Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

If for the ellipse

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

and the hyperbola

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

their foci coincide, then the value of

b^{2}

will be

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

θ_{1} & θ_{2}

are the parameters of the extremities of a chord through

(a e, 0)

of a hyperbola

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

, then

\tan (\frac{θ_{1}}{2}) \tan (\frac{θ_{2}}{2}) =

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

A hyperbola having the transverse axis of length

\sqrt{2}

units has the same foci as that of ellipse

3 x^{2} + 4 y^{2} = 12

, then its equation is

HARD

Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

A point

P

is taken on the right half of the hyperbola

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

having its foci as

S_{1}

and

S_{2} .

If the internal angle bisector of the angle

∠ S_{1} P S_{2}

cuts the

x

-axis at the point

Q (α, 0)

, then

α \in

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

The locus of the foot of perpendicular drawn from the focus of the hyperbola

\frac{x^{2}}{9} - \frac{y^{2}}{4} = 1,

to any arbitrary tangent of the hyperbola, is

HARD

Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

A ray emanating from the point $(5, 0)$ is incident on the hyperbola $9 x^{2} - 16 y^{2} = 144$ at the point $P$ with abscissa $8$ , then the equation of the reflected ray after first reflection is ( $P$ lies in the first quadrant)

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

Four points are such that the line joining any two points is perpendicular to the line joining other two points. If three points out of these lie on a rectangular hyperbola, then the fourth point will lie on.

HARD

Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

If the product of the slopes of the tangents drawn from an external point

P

to the hyperbola

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

is a constant

k^{2}

, then the locus of

P

Tangent at any point P on the hyperbola x29-y24=1 intersects the asymptotes at points A and B, if C is the centre of the hyperbola, then area of Δ ABC is

Important Questions on Hyperbola

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Tangent at any point $P$ on the hyperbola $\frac{x^{2}}{9} - \frac{y^{2}}{4} = 1$ intersects the asymptotes at points $A$ and $B,$ if $C$ is the centre of the hyperbola, then area of $Δ ABC$ is