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

HARD

JEE Main

IMPORTANT

Earn 100

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.

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 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

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

MEDIUM

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 51

Show that the locus of the foot of the perpendicular from the focus on the tangent at any point of the parabola is the tangent at the vertex.

EASY

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 60

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

y^{2} = 4 a x

at the extremities of a focal chord.

EASY

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 63

If the sum of the tangents of the base angles of a triangle described on a given base be constant, show that the locus of its vertex is a parabola.

EASY

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 64

Prove that the locus of the centres of equilateral triangles inscribed in the parabola

y^{2} = 4 a x

is

9 y^{2} = 4 a (x - 8 a) .

EASY

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 71

If the tangents at the points

P

and

Q

on the parabola

y^{2} = 4 a x

meet at

R

and

S

is its focus, prove that

S R^{2} = S P . S Q .

EASY

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 78

The eccentricity of an ellipse is

\frac{1}{2}

and the distance between its foci is

4

units. If the major and minor axes of the ellipse are respectively along the

x

and

y

axes, find the equation of the ellipse.

MEDIUM

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 79

Find the equation to the ellipse (referred to its axes as the axes of

x

and

y

respectively) which passes through the point

(- 3, 1)

and has eccentricity

\sqrt{\frac{2}{5}}