The locus of the mid-points of the chords of the hyperbola x^2-y^2=4, which touch the para…

Question

The locus of the mid-points of the chords of the hyperbola $x^2-y^2=4$, which touch the parabola $y^2=8x$, is

Accepted Answer

Chord with midpoint $(h,k)$: $hx-ky=h^2-k^2$, i.e. $y=\dfrac hkx-\dfrac{h^2-k^2}k$.

Tangency to $y^2=8x$ ($A=2$): $c=\dfrac Am\Rightarrow-\dfrac{h^2-k^2}k=\dfrac{2k}h\Rightarrow-h(h^2-k^2)=2k^2\Rightarrow h^3=k^2(h-2).$

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

Solutions · 1