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

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

Answer 1

Answer: (B)

Let mid point of the chord is $(h,k)$ equation of chord of $x^2-y^2=a^2$ is
$T=S,$
$\Rightarrow hx-ky=h^2-k^2$
$y=\frac{h}{k}x-\left(\frac{h^2-k^2}{k}\right)$.....(1)
(1) is tangent of $y^2=4ax$
condition of tangencys $c=\frac{a}{m}$
$\Rightarrow -\frac{h^2-k^2}{k}=\frac{a}{\left(\frac{h}{k}\right)}\Rightarrow -h^2+k^2=\frac{k^{2}a}{h}$
$\Rightarrow -h^3+h{k}^2=k^{2}a\Rightarrow k^2(h-a)=h^3$
Locus, $y^2(x-a)=x^3$