Beta Question Bank for Engineering: Mathematics>Chapter 18 - Parabola>EXERCISE-4>Q 20

HARD

JEE Main/Advance

IMPORTANT

Earn 100

Two perpendicular straight lines through the focus of the parabola $y^{2} = 4 a x$ meet its directrix in $T & T^{'}$ respectively. Show that the tangents to the parabola parallel to the perpendicular lines intersect in the mid point of $T T^{'}$ .

Important Questions on Parabola

HARD

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 18 - Parabola>EXERCISE-4>Q 21

Find the condition on

' a' & 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 .

HARD

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 18 - Parabola>EXERCISE-4>Q 22

T P

and

T Q

are tangents to the parabola and the normal at

P

and

Q

meet at a point

R

on the curve. Prove that the centre of the circle circumscribing the triangle

T P Q

lies on the parabola

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

MEDIUM

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 18 - Parabola>EXERCISE-4>Q 23

Let

S

is the focus of the parabola

y^{2} = 4 a x

and

X

the foot of the directrix,

P P'

is a double ordinate of the curve and

P X

meets the curve again in

Q .

Prove that

P' Q

passes through focus.

HARD

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 18 - Parabola>EXERCISE-4>Q 25

If

(x_{1}, y_{1}), (x_{2}, y_{2})

and

(x_{3}, y_{3})

be three points on the parabola

y^{2} = 4 a x

and the normals at these points meet in a point, then prove that

\frac{x_{1} - x_{2}}{y_{3}} + \frac{x_{2} - x_{3}}{y_{1}} + \frac{x_{3} - x_{1}}{y_{2}} = 0

HARD

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 18 - Parabola>EXERCISE-4>Q 26

A variable chord joining points

P (t_{1})

and

Q (t_{2})

of the parabola

y^{2} = 4 a x

subtends a right angle at a fixed point

t_{0}

of the curve. Show that it passes through a fixed point. Also find the co-ordinates of the fixed point.

HARD

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 18 - Parabola>EXERCISE-4>Q 27

Show that a circle circumscribing the triangle formed by three co-normal points passes through the vertex of the parabola and its equation is,

2 (x^{2} + y^{2}) - 2 (h + 2 a) x - k y = 0,

where

(h, k)

is the point from where three concurrent normals are drawn.

HARD

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 18 - Parabola>EXERCISE-4>Q 28

If tangent drawn at a point

(t^{2}, 2 t)

on the parabola

y^{2} = 4 x

is same as the normal drawn at a point

(\sqrt{5} \cos ϕ, 2 \sin ϕ)

on the ellipse

4 x^{2} + 5 y^{2} = 20

. Find the values of

t & ϕ

.