91Ó°ÊÓ

Perpendiculars are drawn from points on the tangent at the vertex on their polars with respect to the parabola \(y^{2}=4 a x\). Show that the locus of the foot of the perpendicular is a circle centre at \((a, 0)\) and radius \(a\).

Show that the locus of poles of all chords of the parabola \(y^{2}=4 a x\) which are at a constant distance \(d\) from the vertex is \(d^{2} y^{2}+4 a^{2}\left(d^{2}-x^{2}\right)=0\).

Show that the locus of poles of the focal chords of the parabola \(y^{2}=4 a x\) is \(x+a=0\).

Prove that the polar of any point on the circle \(x^{2}+y^{2}-2 a x-3 a^{2}=0\) with respect to the circle \(x^{2}+y^{2}+2 a x-3 a^{2}=0\) touches the parabola \(y^{2}=4 a x\).

Prove that the length of the chord of contact of the tangents drawn from the point \(\left(x_{1}, y_{1}\right)\) to the parabola \(y^{2}=4 a x\) is \(\frac{1}{a} \sqrt{y_{1}^{2}+4 a^{2}}\left(y_{1}^{2}-4 a x_{1}\right)\). Hence show that one of the triangles formed by these tangents and their chord of contact is \(\frac{1}{2 a}\left(y_{1}^{2}-4 a x_{1}\right)^{3 / 2}\)

Prove that the tangent to a parabola and the perpendicular to it from its focus meet on the tangent at the vertex.

Show that a portion of a tangent to a parabola intercepted between directrix and the curve subtends a right angle at the focus.

Chords of a parabola are drawn through a fixed point. Show that the locus of the middle points is another parabola.

Show that the locus of the middle points of a system of parallel chords of a parabola is a line which is parallel to the axis of the parabola.

Show that the locus of the midpoints of chords of the parabola which subtends a constant angle \(\alpha\) at the vertex is \(\left(y^{2}-2 a x-8 a^{2}\right)^{2} \tan ^{2} \alpha=16 a^{2}\left(4 a x-y^{2}\right)\).