Question

Tangents are drawn from a point $P$ to the parabola $y^2=4ax$. If the chord of contact of the parabola is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, then the locus of $P$ is

• A. $a^2{x}^2=a^4-y^2{b}^2$
• B. $4a^2{x}^2=4a^4-y^2{b}^2$
• C. $4a^2{x}^2=a^4-y^2{b}^2$
• D. $a^2{x}^2=4a^4-y^2{b}^2$

Answer 1

The correct option is B $4a^2{x}^2=4a^4-y^2{b}^2$

Let the coordiates of point $P$ be $(h,k)$
then the equation of the chord of contact of the parabola is
$yk=2a(x+h)$
$\Rightarrow y=\frac{2a}{k}x+\frac{2ah}{k}\cdots \left(1\right)$
Since equation $\left(1\right)$ is a tangent to the hyperbola,
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
So,
$c^2=a^2{m}^2-b^2\Rightarrow {\left(\frac{2ah}{k}\right)}^2=a^2{\left(\frac{2a}{k}\right)}^2-b^2\Rightarrow 4a^2{h}^2=4a^4-k^2{b}^2$

Hence the locus of $P(h,k)$ is
$4a^2{x}^2=4a^4-y^2{b}^2$