MEDIUM

Earn 100

The equation of a line passing through the centre of a rectangular hyperbola is $x - y - 1 = 0 .$ If one of its asymptotes is $3 x - 4 y - 6 = 0,$ the equation of the other asymptote is

50% studentsanswered this correctly

Important Questions on Hyperbola

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

MEDIUM

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

The equations of the asymptotes of the hyperbola

x y + 3 x - 2 y - 10 = 0

are

MEDIUM

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

The equation of the asymptotes of the hyperbola

2 x^{2} + 5 x y + 2 y^{2} - 11 x - 7 y - 4 = 0

MEDIUM

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

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 _____.

HARD

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

The length of the latus rectum of the rectangular hyperbola

x y = 32

EASY

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

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>Rectangular Hyperbola Having Coordinate Axes as Asymptotes

The combined equation of the asymptotes of the hyperbola

2 x^{2} + 5 x y + 2 y^{2} + 4 x + 5 y = 0

EASY

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

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

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 asymptotes of the hyperbola

x y = h x + k y

are

MEDIUM

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

If four points to be taken on a rectangular hyperbola such that the chord joining any two is perpendicular to the chord joining the other two and if

α, β, γ, δ

be the inclination to either asymptote of the straight line joining these points to the centre. Then,

\tan α \tan β \tan γ \tan δ

is equal to

HARD

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

The length of the latusrectum of the hyperbola

x y - 3 x - 3 y + 7 = 0

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

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 -

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

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

The distance between the directrices of the hyperbola

x = 8 \sec θ, y = 8 t a n θ

The equation of a line passing through the centre of a rectangular hyperbola is x-y-1=0. If one of its asymptotes is 3x-4y-6=0, the equation of the other asymptote is

Important Questions on Hyperbola

Learn and Prepare for any exam you want !

The equation of a line passing through the centre of a rectangular hyperbola is $x - y - 1 = 0 .$ If one of its asymptotes is $3 x - 4 y - 6 = 0,$ the equation of the other asymptote is