The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 11 - The Hyperbola>EXAMPLES XXXVII>Q 5

HARD

JEE Advanced

IMPORTANT

Earn 100

Tangents are drawn to a hyperbola from any point on one of the branches of the conjugate hyperbola. Show that their chord of contact will touch the other branch of the conjugate hyperbola.

Important Questions on The Hyperbola

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 11 - The Hyperbola>EXAMPLES XXXVII>Q 6

A straight line is drawn parallel to the conjugate axis of a hyperbola to meet it and the conjugate hyperbola in the points

P

and

Q

respectively. Show that the tangents at

P

and

Q

meet on the curve

\frac{y^{4}}{b^{4}} (\frac{y^{2}}{b^{2}} - \frac{x^{2}}{a^{2}}) = \frac{4 x^{2}}{a^{2}}

and that the normals meet on the axis of

x

.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 11 - The Hyperbola>EXAMPLES XXXVII>Q 7

From a point

G

on the transverse axis

G L

is drawn perpendicular to the asymptote, and

G P

a normal to the curve at

P

. Prove that

L P

is parallel to the conjugate axis.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 11 - The Hyperbola>EXAMPLES XXXVII>Q 8

Find the asymptotes of the curve

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

, and find the general equation of all hyperbolas having the same asymptotes.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 11 - The Hyperbola>EXAMPLES XXXVII>Q 9

Find the equation of the hyperbola, whose asymptotes are the straight lines

x + 2 y + 3 = 0

,and

3 x + 4 y + 5 = 0

, and which passes through the point

(1, - 1)

. Also write down the equation of the conjugate hyperbola.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 11 - The Hyperbola>EXAMPLES XXXVII>Q 10

In a rectangular hyperbola, prove that

C P

and

C D

are equal, and are inclined to the axis at angles which are complementary.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 11 - The Hyperbola>EXAMPLES XXXVII>Q 11

C

is the centre of the hyperbola

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

and the tangent at any point

P

meets the asymptotes in the points

Q

and

R

. Prove that the equation to the locus of the centre of the circle circumscribing the triangle

C Q R

is

4 (a^{2} x^{2} - b^{2} y^{2}) = {(a^{2} + b^{2})}^{2}

.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 11 - The Hyperbola>EXAMPLES XXXVII>Q 12

A series of hyperbolas are drawn having a common transverse axis of length

2 a

. Prove that the locus of a point

P

on each hyperbola, such that its distance from the transverse axis is equal to its distance from an asymptote, is the curve

{(x^{2} - y^{2})}^{2} = 4 x^{2} (x^{2} - a^{2})

.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 11 - The Hyperbola>EXAMPLES XXXVIII>Q 1

Prove that the foci of the hyperbola

x y = \frac{a^{2} + b^{2}}{4}

are given by

x = y = \pm \frac{a^{2} + b^{2}}{2 a}

.