Question

If the chord of contact of the tangents from a point $P$ to the parabola $y^2=4ax$ touches the parabola $x^2=4by$, then the locus of $P$ is

• A. $xy=-2ab$
• B. $xy=ab$
• C. $xy=2ab$
• D. $xy=-ab$

Answer 1

The correct option is A $xy=-2ab$

Let $P\equiv (h,k)$
Equation of chord of contact for parabola $y^2=4ax$ is
$yk=2a(x+h)$
$\Rightarrow y=\frac{2ax}{k}+\frac{2ah}{k}\cdots \left(1\right)$
Equation of tangent for $x^2=4by$ will be of form $y=mx-b{m}^2\cdots \left(2\right)$

As equation $\left(1\right)$ and $\left(2\right)$ represent the same line, so,
$m=\frac{2a}{k}$ and
$\frac{2ah}{k}=-b{m}^2$
$\Rightarrow \frac{2ah}{k}=-b{\left(\frac{2a}{k}\right)}^2$
$\Rightarrow hk=-2ab$
Hence, the locus of the point $P$ will be $xy=-2ab$