Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 5 - Parabola>EXERCISE ON LEVEL-I>Q 13

EASY

JEE Main/Advanced

IMPORTANT

Earn 100

Prove that the circles described on the focal radii as diameters touch the tangent at the vertex.

Important Questions on Parabola

HARD

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 5 - Parabola>EXERCISE ON LEVEL-I>Q 14

Show that

l x + m y + n = 0

is a normal to the parabola

y^{2} = 4 a x

if

a l^{3} + 2 a l m^{2} + m^{2} n = 0

.

HARD

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 5 - Parabola>EXERCISE ON LEVEL-I>Q 15

Find the locus of the middle points of chords of the parabola which are such that the normal at their extremities meet on the parabola.

EASY

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 5 - Parabola>EXERCISE ON LEVEL-I>Q 16

T P

and

T Q

are tangents to a parabola and

p_{1}, p_{2}, p_{3}

are the lengths of the perpendiculars from

P, T, Q

respectively on any tangent to the curve. Prove that

p_{1} p_{3} = p_{2}^{2}

.

HARD

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 5 - Parabola>EXERCISE ON LEVEL-I>Q 17

Find the condition on

a

and

b

so that the two tangents drawn to the parabola

y^{2} = 4 a x

from a point are normals to the parabola

x^{2} = 4 b y

.

EASY

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 5 - Parabola>EXERCISE ON LEVEL-I>Q 18

On the parabola

y^{2} = 4 a x

three point

P, Q, R

are taken so that their ordinates are in geometrical progression. Prove that the tangents at

P

and

R

meet on abscissa of

Q

.

MEDIUM

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 5 - Parabola>EXERCISE ON LEVEL-I>Q 19

Prove that three normals can be drawn from the point $(c, 0)$ to the parabola

y^{2} = x

if

c > \frac{1}{2}

and then one of the normals is always the axis of the parabola

EASY

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 5 - Parabola>EXERCISE ON LEVEL-I>Q 21

Show that the length of the tangent to the parabola

y^{2} = 4 a x

is intercepted between its point of contact and the axis of the parabola is bisected by the tangent at the vertex

HARD

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 5 - Parabola>EXERCISE ON LEVEL-I>Q 22

Prove that the normal at

(a m^{2}, - 2 a m)

to the parabola

y^{2} = 4 a x

intersects the parabola again at an angle

\tan^{- 1} (\frac{m}{2}), m > 0 .