Question

If chords of the hyperbola $x^2-y^2=a^2$ touch the parabola $y^2=4ax$. Then the locus of the middle points of these chords is the curve is :

• A. $y^2=(x-a)x^3$
• B. $y^2(x-a)=x^3$
• C. $x^2(x-a)=y^3$
• D. None of these

Answer 1

The correct option is B $y^2(x-a)=x^3$

Equation of chord of hyperbola $x^2-y^2=a^2$ with mid-point as $(h,k)$ is given by

$xh-yk=h^2-k^2\Rightarrow y=\frac{h}{k}x-\frac{(h^2-k^2)}{k}$

This will touch the parabola $y^2=4ax$ if

$-\frac{(h^2-k^2)}{k}=\frac{a}{h/k}\Rightarrow k^2(h-a)=h^3$

$\therefore$ The locus is $y^2(x-a)=x^3$