S L Loney Solutions for Chapter: The Parabola (Continued), Exercise 1: EXAMPLES XXIX

Find the locus of the middle point of chord of the parabola $y^2=4ax$ which passes through the fixed point $\left(h, k\right)$.

Find the locus of the middle points of the chords of the parabola which are normal to the curve.

Find the locus of the middle points of the chords of the parabola which subtend a constant angle $\theta$ at the vertex.

Find the locus of the middle points of chords of the parabola $y^2=4ax$ which are of given length $l.$

Find the locus of the middle points of chords of the parabola which are such that the normal at their extremities meet on the parabola.

Through each point of the straight line $x=my+h$ a chord of the parabola $y^2=4ax$ is drawn, which is bisected at the point. Prove that it always touches the parabola ${\left(y+2am\right)}^2=8a\left(x-h\right)$.

Two parabolas have the same axis and tangents are drawn to the second from points on the first. Prove that the locus of the middle points of the chords of contact with the second parabola all lie on a fixed parabola.

Prove that the locus of the feet of the perpendicular drawn from the vertex of the parabola $y^2=4ax$ upon chords, which subtend an angle $45°$ at the vertex, is the curve $r^2-24ar\cos \theta +16a^2\cos 2\theta =0$. (where $r$ is the perpendicular distance of the chord from the vertex)