Question

If the chords of the rectangular hyperbola $x^2-y^2=a^2$ touch the parabola $y^2=4ax$, then the locus of their mid-points is:

• A. $x^2(y-a)=y^3$
• B. $y^2(x-a)=x^3$
• C. $x(y^2-a)=y$
• D. $y(x^2-a)=x$

Answer 1

Answer: (B)

$y^2(x-a)=x^3$

Let $P(h,k)$ be the midpoint of the chord,
Thus equation of chord to the hyperbola $x^2-y^2=a^2$ is,
$hx-ky=h^2-k^2\Rightarrow y=\frac{h}{k}x+k-\frac{h^2}{k}$
Now given this line touches parabola $y^2=4ax$, so using condition of tangency,
$c=\frac{a}{m}\Rightarrow k-\frac{h^2}{k}=a\frac{k}{h}$
$\Rightarrow k^2(h-a)=h^3$
Therefore, locus of $P(h,k)$ is, $y^2(x-a)=x^3$
Hence, option 'B' is correct.