Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 2

HARD

JEE Main

IMPORTANT

Earn 100

Find the vertex, focus, axis, directrix, and latus-rectum of the following parabola

$y^{2} + 4 x + 4 y - 3 = 0$

Important Questions on Conic Section

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 3

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 .

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 4

If the normals to the parabola

y^{2} = 4 a x

at three points

P, Q

and

R

meet at

A

and

S

be the focus, prove that

S P . S Q . S R

= a . S A^{2},

where

a > 0

.

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 5

The normals to the parabola

y^{4} = 4 a x

from a point

P

meet the axis at

A, B, C

. If

B

is the middle point of

A C

, find the locus of

P

.

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 6

Show that the equation of the chord of the parabola

y^{2} = 4 a x

through the points

(x_{1}, y_{1})

and

(x_{2}, y_{2})

on it is

(y - y_{1}) (y - y_{2}) = y^{2} - 4 a x .

MEDIUM

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 7

Prove that the middle point of the intercept made on a tangent to a parabola by the tangents at two points

P

and

Q

lies on the tangent which is parallel to

P Q

.

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 8

Show that the orthocentre of the triangle formed by the tangents at the points

t_{1}, t_{2}, t_{3}

to the parabola

y^{2} = 4 a x

is

$(- a, a (t_{1} + t_{2} + t_{3} + t_{1} t_{2} t_{3}))$

MEDIUM

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 9

Show that any three tangents to a parabola whose slopes are in

H . P .

enclose a triangle of constant area.

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 10

Prove that the circumcircle of the triangle formed by any three tangents to the parabola

y^{2} = 4 a x

passes through its focus.