Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 6 - Ellipse>INTRODUCTORY EXERCISE 6.2>Q 4

HARD

JEE Main/Advanced

IMPORTANT

Earn 100

The tangent and normal at any point $P$ of an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ cut its major axis in point $Q$ and $R$ respectively. If $Q R = a$ prove that the eccentric angle of the point $P$ is given by $e^{2} \cos^{2} ϕ + \cos ϕ - 1 = 0$

Important Questions on Ellipse

MEDIUM

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 6 - Ellipse>INTRODUCTORY EXERCISE 6.3>Q 1

The angle between the pair of tangents drawn from the point

(1, 2)

to the ellipse

3 x^{2} + 2 y^{2} = 5

is

MEDIUM

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 6 - Ellipse>INTRODUCTORY EXERCISE 6.3>Q 2

If the chords of contact of tangent from two points

(x_{1}, y_{1})

and

(x_{2}, y_{2})

to the ellipse

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

are at right angles, then

\frac{x_{1} x_{2}}{y_{1} y_{2}}

is equal to

MEDIUM

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 6 - Ellipse>INTRODUCTORY EXERCISE 6.3>Q 3

From the point

(λ, 3)

tangents are drawn to

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

and are perpendicular to each other than

λ

is :

MEDIUM

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 6 - Ellipse>INTRODUCTORY EXERCISE 6.3>Q 4

The eccentric angle of one end of a diameter of

x^{2} + 3 y^{2} = 3

is

π / 6

, then the eccentric angle of the other end will be :

MEDIUM

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 6 - Ellipse>INTRODUCTORY EXERCISE 6.3>Q 5

The pole of the straight line

x + 4 y = 4

with respect to the ellipse

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

is

EASY

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 6 - Ellipse>INTRODUCTORY EXERCISE 6.3>Q 6

The locus of the mid-points of a focal chord of the ellipse

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

is :

HARD

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 6 - Ellipse>INTRODUCTORY EXERCISE 6.3>Q 8

The centre of the ellipse

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

is :

HARD

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 6 - Ellipse>Multiple Choice I>Q 10

If the polar with respect to

y^{2} = 4 a x

touches the ellipse

\frac{x^{2}}{α^{2}} + \frac{y^{2}}{β^{2}} = 1,

the locus of its pole is