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

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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 $P$ is

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Important Questions on Hyperbola

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

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>Properties of Hyperbola

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>Properties of Hyperbola

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

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?

EASY

Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

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>Properties of Hyperbola

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>Properties of Hyperbola

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.

HARD

Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

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>Properties of Hyperbola

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>Properties 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:

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Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

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

EASY

Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

If the line

2 x + \sqrt{6} y = 2

touches the hyperbola

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

then the point of contact is

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

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

HARD

Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

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>Properties of Hyperbola

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

EASY

Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

Let a tangent be drawn to the ellipse

\frac{x^{2}}{27} + y^{2} = 1

(3 \sqrt{3} \cos θ, \sin θ)

where

θ \in (0, \frac{π}{2})

. Then the value of

θ

such that the sum of intercepts on axes made by this tangent is minimum is equal to :

EASY

Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

The straight line

x + y = \sqrt{2} p

will touch the hyperbola

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

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Mathematics>Coordinate Geometry>Hyperbola>Properties of Hyperbola

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>Properties of Hyperbola

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>Properties of Hyperbola

The equation of a tangent to the hyperbola,

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

parallel to the line

x - y = 2,

If the sum of slopes of concurrent normals to the curve xy=4 is equal to the sum of ordinates of conormal points then locus of P is

Important Questions on Hyperbola

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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 $P$ is