Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 24

MEDIUM

JEE Main

IMPORTANT

Earn 100

Prove that the circle described on any focal chord of a parabola as diameter touches the directrix.

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 25

A circle on any focal chord of a parabola as diameter cuts the curve again in

P

and

Q

. Show that

P Q

passes through a fixed point.

MEDIUM

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 26

Circles are described on any two focal chords of a parabola, prove that their common chord passes through the vertex.

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 27

Show that the tangent at one extremity of a focal chord of a parabola is parallel to the normal at other extremity.

MEDIUM

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 28

The tangents and normals at the ends of a focal chord of a parabola meet in

P

and

M

respectively. Show that

P M

is parallel to the axis.

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 29

Show that the locus of the middle point of chords of the parabola

y^{2} = 4 a x

which subtend a right angle at the vertex is

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

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 30

Show that all chords of a parabola which subtend a right angle at the vertex pass through a fixed point on the axis of the curve.

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 31

Show that if $r_{1}$ and $r_{2}$ be the lengths of perpendicular chords of a parabola drawn through the vertex, then

${(r_{1} r_{2})}^{\frac{4}{3}} = 16 a^{2} (r_{1}^{\frac{2}{3}} + r_{2}^{\frac{2}{3}})$

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 32

Show that the equation of the circle described on the chord intercepted by the parabola

y^{2} = 4 a x

on the line

y = m x + c

as diameter is

$m^{2} (x^{2} + y^{2}) + 2 x (m c - 2 a) - 4 a m y + 4 a m c + c^{2} = 0 .$