[Math] Prove that a triangle can be inscribed in the hyperbola $xy=c^2$ whose sides touch the parabola $y^2=4ax$.

Question

Prove that a triangle can be inscribed in the hyperbola $xy=c^2$ whose sides touch the parabola $y^2=4ax$.

I have no idea how to start this problem. In my first attempt I assumed any three points on the given rectangular hyperbola and wrote down the equations of each of the three chords. Then I tried proving that each of the three sides are tangents to the given parabola…then I put the value of $x$ from the equation of the chord in the equation of the parabola and tried to prove that the discriminant of the quadratic is equal to zero…but I am unable to proceed after that.

Need help…thanks!!

Answer 1

HINT.

From any point $A$ of the hyperbola, in the branch intersecting the parabola but external to it (see diagram), draw tangents $AD$ and $AF$ to the parabola, which intersect the other branch of the hyperbola at $B$ and $C$. Check that line $BC$ is also tangent to the parabola.