Consider the hyperbola H : x 2 - y 2 = 1 and a circle S with center N x 2 , 0 . Suppose that H and S touch each other at point P x 1 , y 1 with x 1 > 1 & y 1 > 0 . The common tangent to H and S at P intersects the x -axis at point M . If l , m is the centroid of the triangle Δ P M N , then the correct expression(s) is (are)

d l d x 1 = 1 + 1 3 x 1 2 for x 1 > 1

EASY

Earn 100

What is the slope of the tangent drawn to the hyperbola $x y = a, (a \neq 0)$ at the point $(a, 1) ?$

50% studentsanswered this correctly

Important Questions on Hyperbola

EASY

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

Let

P

be the point of intersection of the common tangents to the parabola

y^{2} = 12 x

and the hyperbola

8 x^{2} - y^{2} = 8

. If

S

and

S^{'}

denote the foci of the hyperbola where

S

lies on the positive

x

-axis then

P

divides

S S^{'}

in a ratio:

HARD

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

2 x - y + 1 = 0

is a tangent to the hyperbola

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

, then which of the following CANNOT be sides of a right angled triangle?

MEDIUM

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

If the line

y = m x + 7 \sqrt{3}

is normal to the hyperbola

\frac{x^{2}}{24} - \frac{y^{2}}{18} = 1

, then a value of

m

is:

MEDIUM

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

Let

P (3, 3)

be a point on the hyperbola,

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

If the normal to it at

P

intersects the

x

-axis at

(9, 0)

and

e

is its eccentricity, then the ordered pair

(a^{2}, e^{2})

is equal to:

EASY

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

If the line

2 x + \sqrt{6} y = 2

touches the hyperbola

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

then the point of contact is

HARD

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

A line parallel to the straight line

2 x - y = 0

is tangent to the hyperbola

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

at the point

(x_{1}, y_{1})

. Then

x_{1}^{2} + 5 y_{1}^{2}

is equal to

EASY

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

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:

MEDIUM

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

The distance between the tangents to the hyperbola

\frac{x^{2}}{20} - \frac{3 y^{2}}{4} = 1

which are parallel to the line

x + 3 y = 7

MEDIUM

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

A hyperbola passes through the point

P (\sqrt{2}, \sqrt{3})

and has foci at

(\pm 2, 0)

. Then the tangent to this hyperbola at

P

also passes through the point

EASY

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

The tangent at an extremity (in the first quadrant) of the latus rectum of the hyperbola $\frac{x^{2}}{4} - \frac{y^{2}}{5} = 1$ , meets the $x$ -axis and $y$ -axis at $A$ and $B$ , respectively. Then $O A^{2} - O B^{2}$ , where $O$ is the origin, equals

HARD

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

If the line

y = m x + c

is a common tangent to the hyperbola

\frac{x^{2}}{100} - \frac{y^{2}}{64} = 1

and the circle

x^{2} + y^{2} = 36,

then which one of the following is true?

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

HARD

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

Let

a

and

b

be positive real numbers such that

a > 1

and

b < a .

Let

P

be a point in the first quadrant that lies on the hyperbola

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

Suppose the tangent to the hyperbola at

P

passes through the point

(1, 0),

and suppose the normal to the hyperbola at

P

cuts off equal intercepts on the coordinate axes. Let

Δ

denote the area of the triangle formed by the tangent at

P

, the normal at

P

and the

x

-axis. If

e

denotes the eccentricity of the hyperbola, then which of the following statements is/are TRUE?

MEDIUM

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

If a hyperbola passes through the point

P (10, 16)

, and it has vertices at

(\pm 6, 0)

, then the equation of the normal to it at

P

, is.

MEDIUM

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

A tangent drawn to hyperbola

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

P (\frac{π}{6})

forms a triangle of area

3 a^{2}

square units, with coordinate axes. If the eccentricity of hyperbola is

e

, then the value of

e^{2} - 9

HARD

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

Equation of a tangent to the hyperbola

5 x^{2} - y^{2} = 5

and which passes through an external point

(2, 8)

HARD

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

Let

P (4, 3)

be a point on the hyperbola

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

If the normal at

P

intersects the

X -

axis at

(16, 0),

then the eccentricity of the hyperbola is

HARD

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

Consider the hyperbola

H : x^{2} - y^{2} = 1

and a circle

S

with center

N (x_{2}, 0) .

Suppose that

H

and

S

touch each other at point

P (x_{1}, y_{1})

with

x_{1} > 1 & y_{1} > 0 .

The common tangent to

H

and

S

P

intersects the

x

-axis at point

M

. If

(l, m)

is the centroid of the triangle

Δ P M N,

then the correct expression(s) is (are)

EASY

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

The equation of a tangent to the hyperbola,

4 x^{2} - 5 y^{2} = 20,

parallel to the line

x - y = 2,

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

What is the slope of the tangent drawn to the hyperbola xy=a, a≠0 at the point a, 1?

Important Questions on Hyperbola

Learn and Prepare for any exam you want !

What is the slope of the tangent drawn to the hyperbola $x y = a, (a \neq 0)$ at the point $(a, 1) ?$