Question

If the chord of the hyperbola $x^2-y^2=a^2$ touch the parabola $y^2=4ax,$ then the locus of the middle point of these chord is _____

• A. y^2\)
• B. y^2\)
• C. y^2\)
• D. y\)

Answer 1

The correct option is A

y^2\)

The equation of the chord of the hyperbola whose middle point is (h,k) is T=$s_1$

$i.e.,xh-yk=h^2-k^2$

$yk=xh-h^2+k^2$

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

We know that hte line y=mx+c will touch the parabola if c=am

Hence $\frac{-(h^2-k^2)}{k}=\frac{a}{\frac{h}{k}}$

$\frac{-(h^2-k^2)}{k}=\frac{ka}{h}$

$h(h^2-k^2)=-a{k}^2$

$\Rightarrow h^3-h{k}^2=-a{k}^2$

$h^3=(h-a)k^2$

Hence, the locus of (h,k) is $x^3=(x-a)y^2$