Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 5 - Parabola>INTRODUCTORY EXERCISE 5.2>Q 5

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IMPORTANT

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If the normals at two points $P$ and $Q$ of a parabola $y^{2} = 4 a x$ intersect at a third point $R$ on the curve, then the product of ordinates of $P$ and $Q$ is

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Important Questions on Parabola

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JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 5 - Parabola>INTRODUCTORY EXERCISE 5.2>Q 7

The set of points on the axis of the parabola

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

from which all the three normals to the parabola are real is :

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JEE Main/Advanced

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Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 5 - Parabola>INTRODUCTORY EXERCISE 5.2>Q 1

Prove that any three tangents to a parabola whose slopes are in harmonic progression enclose a triangle of constant area.

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JEE Main/Advanced

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Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 5 - Parabola>INTRODUCTORY EXERCISE 5.2>Q 2

A chord of parabola

y^{2} = 4 a x

subtends a right angle at the vertex. Find the locus of the point of intersection of tangents at its extremities.

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JEE Main/Advanced

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Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 5 - Parabola>INTRODUCTORY EXERCISE 5.2>Q 3

Find the equation of the normal to the parabola

y^{2} = 4 x

which is :
(a) parallel to the line

y = 2 x - 5

(b) Perpendicular to the line

2 x + 6 y + 5 = 0

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JEE Main/Advanced

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Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 5 - Parabola>INTRODUCTORY EXERCISE 5.2>Q 4

The ordinates of points

P

and

Q

on the parabola

y^{2} = 12 x

are in the ratio

1 : 2

. Find the locus of the point of intersection of the normals to the parabola at

P

and

Q

.

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JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 5 - Parabola>INTRODUCTORY EXERCISE 5.2>Q 5

The normals at

P, Q, R

on the parabola

y^{2} = 4 a x

meet in a point on the line

y = c

. Prove that the sides of the triangle

P Q R

touch the parabola

x^{2} = 2 c y

.

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JEE Main/Advanced

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Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 5 - Parabola>INTRODUCTORY EXERCISE 5.2>Q 6

The normals are drawn from

(2 λ, 0)

to the parabola

y^{2} = 4 x

. Show that $\lambda$ must be greater than

1

. One normal is always the

x

-axis. Find

λ

for which the other two normals are perpendicular to each other.

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Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 5 - Parabola>INTRODUCTORY EXERCISE 5.3>Q 1

If

m_{1}, m_{2}

are slopes of the two tangents that are drawn from

(2, 3)

to the parabola

y^{2} = 4 x

then the value of

\frac{1}{m_{1}} + \frac{1}{m_{2}}

is: