Question

Prove that the locus of the point of intersection of the tangents at the ends of the normal chords of the hyperbola `x^2-y^2=a^2` is `a^2(y^2-x^2)=4x^2y^2dot`

Answer 1

Normal at a point `(a sec theta, a tan theta)` is
`x cos theta+y cot theta=2a`
If `P(x_(1),y_(1))` is the point of intersection of the tangents at the ends of normal chord (1), then (1) must be the chord of contact of P(h, k) whose equation is given by
`hx-ky=a^(2)" (2)"`
Comparing (1) and (2) and eliminating `theta`, we get
`(a^(2))/(4h^(2))-(a^(2))/(4k^(2))=1`
Hence, the locus is
`(1)/(x^(2))-(1)/(y^(2))=(4)/(a^(2))`