Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 7 - Hyperbola>INTRODUCTORY EXERCISE 7.1>Q 5

MEDIUM

JEE Main/Advanced

IMPORTANT

Earn 100

Find the eccentricity of the conjugate hyperbola $x^{2} - 3 y^{2} = 1$ .

Important Questions on Hyperbola

HARD

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 7 - Hyperbola>INTRODUCTORY EXERCISE 7.1>Q 6

For what value of

λ,

does the line

y = 3 x + λ

touch the hyperbola

9 x^{2} - 5 y^{2} = 45 ?

MEDIUM

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 7 - Hyperbola>INTRODUCTORY EXERCISE 7.2>Q 1

A common tangent to

9 x^{2} - 16 y^{2} = 144

and

x^{2} + y^{2} = 9

, is

HARD

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 7 - Hyperbola>INTRODUCTORY EXERCISE 7.2>Q 2

The tangents from

(1, 2 \sqrt{2})

to the hyperbola

16 x^{2} - 25 y^{2} = 400

include between them an angle equal to

EASY

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 7 - Hyperbola>INTRODUCTORY EXERCISE 7.2>Q 6

P

is a point on the hyperbola

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

is the foot of the perpendicular from

P

on the transverse axis. The tangent to the hyperbola at

P

meets the transverse axis at

T

. If

O

is the centre of the hyperbola, then

O T \cdot O N

is equal to

MEDIUM

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 7 - Hyperbola>INTRODUCTORY EXERCISE 7.2>Q 4

Chords of the hyperbola,

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

touch the parabola,

y^{2} = 4 a x

. Prove that the locus of their middle points is the curve,

y^{2} (x - a) = x^{3}

EASY

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 7 - Hyperbola>INTRODUCTORY EXERCISE 7.3>Q 3

If

H \equiv \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} - 1 = 0, C \equiv \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} + 1 = 0

and

A \equiv \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 0

then

H, A

and

C

are in :

EASY

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 7 - Hyperbola>INTRODUCTORY EXERCISE 7.3>Q 4

The angle between the asymptotes of

\frac{x^{2}}{4} - \frac{y^{2}}{9} = 1

is equal to :

EASY

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 7 - Hyperbola>INTRODUCTORY EXERCISE 7.3>Q 5

If

e

and

e_{1}

are the eccentricities of the hyperbolas

x y = c^{2}

and

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

, then

{(e + e_{1})}^{2}

is equal to :