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The equation of the director circle to the rectangular hyperbola $x^{2} - y^{2} = a^{2}$ is

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Important Questions on Hyperbola

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Mathematics>Coordinate Geometry>Hyperbola>Rectangular Hyperbola Having Coordinate Axes as Asymptotes

If any tangent to the parabola

x^{2} = 4 y

intersects the hyperbola

x y = 2

at two points

P

and

Q,

then the mid-point of line segment

P Q

lies on a parabola with axis along

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Rectangular Hyperbola Having Coordinate Axes as Asymptotes

Tangents are drawn to the hyperbola

4 x^{2} - y^{2} = 36

at the points

P

and

Q

. If these tangents intersect at the point

T (0, 3)

then the area (in sq. units) of

Δ P T Q

is:

EASY

Mathematics>Coordinate Geometry>Hyperbola>Rectangular Hyperbola Having Coordinate Axes as Asymptotes

If the pole of the line

3 x - 16 y + 48 = 0

with respect to the hyperbola

9 x^{2} - 16 y^{2} = 144

(α, β),

then

α - β =

HARD

Mathematics>Coordinate Geometry>Hyperbola>Rectangular Hyperbola Having Coordinate Axes as Asymptotes

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

EASY

Mathematics>Coordinate Geometry>Hyperbola>Rectangular Hyperbola Having Coordinate Axes as Asymptotes

The equation of the chord of the hyperbola

25 x^{2} - 16 y^{2} = 400

that is bisected at point

(5, 3)

HARD

Mathematics>Coordinate Geometry>Hyperbola>Rectangular Hyperbola Having Coordinate Axes as Asymptotes

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

HARD

Mathematics>Coordinate Geometry>Hyperbola>Rectangular Hyperbola Having Coordinate Axes as Asymptotes

The length of the transverse axis of the rectangular hyperbola

x y = 18

HARD

Mathematics>Coordinate Geometry>Hyperbola>Rectangular Hyperbola Having Coordinate Axes as Asymptotes

The tangent to the hyperbola xy = c² at the point P intersects the x - axis at T and the y - axis at

T^{'}

. The normal to the hyperbola at P intersects the x - axis at N and the y - axis at

N^{'}

. If the areas of the triangles PNT and

{PN}^{'} T^{'}

are

Δ and Δ^{'}

respectively, then

\frac{1}{Δ} + \frac{1}{Δ^{'}}

HARD

Mathematics>Coordinate Geometry>Hyperbola>Rectangular Hyperbola Having Coordinate Axes as Asymptotes

If circle

x^{2} + y^{2} + 2 g x + 2 f y + k = 0

, intersect hyperbola

x y = c^{2}

at four points

(x_{i}, y_{i}), i = 1, 2, 3, 4,

then

HARD

Mathematics>Coordinate Geometry>Hyperbola>Rectangular Hyperbola Having Coordinate Axes as Asymptotes

A circle cuts rectangular hyperbola

x y = 1

in the points

(x_{r}, y_{r}), r = 1,2, 3, 4,

then

EASY

Mathematics>Coordinate Geometry>Hyperbola>Rectangular Hyperbola Having Coordinate Axes as Asymptotes

If the line

a x + b y - c = 0

is normal to the curve

x y = 1,

then

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Rectangular Hyperbola Having Coordinate Axes as Asymptotes

P M

and

P N

are the perpendiculars from any point on a rectangular hyperbola to its asymptotes. If

Q

divides

M N

in the ratio

3 : 1,

then the locus of

Q

is -

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Rectangular Hyperbola Having Coordinate Axes as Asymptotes

If distance between directrices of a rectangular hyperbola is

10,

then distance between its foci will be

HARD

Mathematics>Coordinate Geometry>Hyperbola>Rectangular Hyperbola Having Coordinate Axes as Asymptotes

If the circle

x^{2} + y^{2} = a^{2}

intersects the hyperbola

x y = c^{2}

in four points

P (x_{1}, y_{1}), Q (x_{2}, y_{2}), R (x_{3}, y_{3})

and

S (x_{4}, y_{4})

, then

HARD

Mathematics>Coordinate Geometry>Hyperbola>Rectangular Hyperbola Having Coordinate Axes as Asymptotes

The normals at three points

P, Q, R

on a rectangular hyperbola

x y = c^{2}

intersect at a point on the curve. Then the centre of the hyperbola is a special point of the triangle

P Q R

, which is

HARD

Mathematics>Coordinate Geometry>Hyperbola>Rectangular Hyperbola Having Coordinate Axes as Asymptotes

The asymptotes of the hyperbola

x y = h x + k y

are

HARD

Mathematics>Coordinate Geometry>Hyperbola>Rectangular Hyperbola Having Coordinate Axes as Asymptotes

If normal to the hyperbola

x y = c^{2}

, at point

t

meets the curve again at a point

t_{1}

, then

t^{3} t_{1} + 1

is equal to

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Rectangular Hyperbola Having Coordinate Axes as Asymptotes

The distance between the directrices of the hyperbola

x = 8 \sec θ, y = 8 t a n θ

HARD

Mathematics>Coordinate Geometry>Hyperbola>Rectangular Hyperbola Having Coordinate Axes as Asymptotes

The tangent and normal to a rectangular hyperbola

x y = c^{2}

at a point cuts off intercepts

a_{1} & a_{2}

on one axis and

b_{1}, b_{2}

on the other, then

a_{1} a_{2} + b_{1} b_{2}

is equal to

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Rectangular Hyperbola Having Coordinate Axes as Asymptotes

Length of the straight line

x - 3 y = 1

intercepted by the hyperbola

x^{2} - 4 y^{2} = 1

The equation of the director circle to the rectangular hyperbola x2-y2=a2 is

Important Questions on Hyperbola

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The equation of the director circle to the rectangular hyperbola $x^{2} - y^{2} = a^{2}$ is