Find the equation and lengths of common tangents to the hyperbolas x 2 a 2 - y 2 b 2 = 1 and y 2 a 2 - x 2 b 2 = 1 a > b

y = ± x ± a 2 - b 2 , a 2 + b 2 2 a 2 - b 2

y = ± x ± a 2 - b 2 , a 2 - b 2 2 a 2 - b 2

y = x a 2 - b 2 , a 2 + b 2 2 a 2 - b 2

y = x ± a 2 - b 2 , a 2 + b 2 2 a 2 - b 2

Find the equation and lengths of common tangents to the hyperbolas x 2 a 2 - y 2 b 2 = 1 and y 2 a 2 - x 2 b 2 = 1 a > b

y = ± x ± a 2 - b 2 , a 2 + b 2 2 a 2 - b 2

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Find the equation and lengths of common tangents to the hyperbolas $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 and \frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1 (a > b)$

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Important Questions on Hyperbola

EASY

Mathematics>Coordinate Geometry>Hyperbola>Basics 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

HARD

Mathematics>Coordinate Geometry>Hyperbola>Basics 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>Basics 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>Basics 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>Basics 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:

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Basics 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

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Basics 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:

HARD

Mathematics>Coordinate Geometry>Hyperbola>Basics of Hyperbola

The total number of points on the curve

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

at which the tangents to the curves are parallel to the line

x = 2 y

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Basics 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:

EASY

Mathematics>Coordinate Geometry>Hyperbola>Basics 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:

EASY

Mathematics>Coordinate Geometry>Hyperbola>Basics of Hyperbola

The straight line

x + y = \sqrt{2} p

will touch the hyperbola

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

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Basics 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>Basics of Hyperbola

Consider a hyperbola

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

. Let the tangent at a point

P (4, \sqrt{6})

meet the

x

-axis at

Q

and latus rectum at

R (x_{1}, y_{1}), x_{1} > 0

. If

F

is a focus of

H

which is nearer to the point

P

, then the area of

Δ Q F R

(in sq. units) is equal to

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Basics 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

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Basics of Hyperbola

The locus of the midpoints of the chord of the circle,

x^{2} + y^{2} = 25

which is tangent to the hyperbola,

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

is :

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Basics of Hyperbola

The point

P (- 2 \sqrt{6}, \sqrt{3})

lies on the hyperbola

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

having eccentricity

\frac{\sqrt{5}}{2} .

If the tangent and normal at $P$ to the hyperbola intersect its conjugate axis at the points

Q

and

R

respectively, then

Q R

is equal to:

HARD

Mathematics>Coordinate Geometry>Hyperbola>Basics of Hyperbola

Let a line

L : 2 x + y = k, k > 0

be a tangent to the hyperbola

x^{2} - y^{2} = 3 .

L

is also a tangent to the parabola

y^{2} = α x,

then

α

is equal to:

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Basics 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.

EASY

Mathematics>Coordinate Geometry>Hyperbola>Basics 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

EASY

Mathematics>Coordinate Geometry>Hyperbola>Basics of Hyperbola

The equation of a tangent to the hyperbola,

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

parallel to the line

x - y = 2,

Find the equation and lengths of common tangents to the hyperbolas x2a2-y2b2=1 and y2a2-x2b2=1 a>b

Important Questions on Hyperbola

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Find the equation and lengths of common tangents to the hyperbolas $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 and \frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1 (a > b)$