Question: If the chord of contact of tangents from a point P to the parabola y<sup>2</sup> = 4ax, touches the ...

Question

If the chord of contact of tangents from a point P to the parabola y² = 4ax, touches the parabola x² = 4by, then the locus of P is a/an –

Answer 1

Let P(h, k) be a point. Then the chord of contact of tangents from P to y² = 4ax is

ky = 2a (x + h)

This touches the parabola x² = 4by. So, it should be of the form

x = my + $\frac{b}{m}$ … (2)

Equation (1) can be re-written as

x = $\left( \frac{k}{2a} \right)$ y – h … (3)

Since Equation (2) and (3) represent the same line.

\ m = $\frac{k}{2a}$ and $\frac{b}{m}$ = –h

Eliminating m from these two equations, we get

2ab = –hk

Hence, the locus of P(h, k) is

xy = –2ab, which is a hyperbola.

Hence (4) is the correct answer.