A circle is described on a focal chord of the parabola $y^{2} = 4 a x$ , as diameter, if $m$ be the tangent of the inclination of the chord with the axis, prove that the equation to the circle is $x^{2} + y^{2} - 2 a x (1 + \frac{2}{m^{2}}) - \frac{4 a y}{m} - 3 a^{2} = 0$

Important Questions on Conic Sections. The Parabola

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 8 - Conic Sections. The Parabola>EXAMPLES XXVIII>Q 24

L O L'

and

M O M'

are two chords of a parabola, passing through a point

O

on its axis. Prove that the radical axis of the circles described on

\frac{4}{m^{2}} = t^{2} + \frac{1}{t^{2}} - 2

and

M M'

as diameters, passes through the vertex of the parabola.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 8 - Conic Sections. The Parabola>EXAMPLES XXVIII>Q 25

A circle and a parabola intersect at four points; show that the algebraic sum of the ordinates of the four points is zero. Also, show that the line joining one pair of these four points and the line joining the other pair are equally inclined to the axis.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 8 - Conic Sections. The Parabola>EXAMPLES XXVIII>Q 26

Circles are drawn through the vertex of the parabola to cut the parabola orthogonally at the other point of intersection. Prove that the locus of the centers of the circles is the curve