S L Loney Solutions for Chapter: The Hyperbola, Exercise 1: EXAMPLES XXXVI

If one axis of a varying central conic be fixed in magnitude and position, prove that the locus of the point of contact of a tangent drawn to it from a fixed point on the other axis is a parabola.

If the ordinate $MP$ of hyperbola be produced to $Q$ so that $MQ$  is equal to either of the focal distance of $P$, prove that the locus of $Q$ is one or other of a pair of parallel straight lines.

Show that the locus of the centre of a circle which touches externally two given circles is a hyperbola.

Given the base of a triangle and the ratio of the tangents of half of the base of the angles, prove that the vertex moves on a hyperbola whose foci are extremities of the base.

Find the locus of the pole of a chord of the hyperbola which subtends a right angle at the vertex.

Prove that the locus of the intersection of tangents to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, which meet at a constant angle $\beta$, is the curve ${\left(x^2+y^2+b^2-a^2\right)}^2=4{\cot }^2\beta \left(a^2{y}^2-b^2{x}^2+a^2{b}^2\right)$.

From points on the circle $x^2+y^2=a^2$ tangents are drawn to the hyperbola $x^2-y^2=a^2$. Prove that the locus of the middle points of the chords of contact is the curve ${\left(x^2-y^2\right)}^2=a^2\left(x^2+y^2\right)$.

Chords of hyperbola are drawn, all passing through the fixed point $(h, k)$. Prove that the locus of their middle points is a hyperbola whose centre is the point $\left(\frac{h}{2}, \frac{k}{2}\right)$ and which is similar to either the hyperbola or its conjugate.