Prove that an infinite number of triangles can be inscribed in the hyperbola xy=c^2 whose sides touch the parabola y^2=4ax.

Question

So I tried writing 3 equations of tangents to the parabola:

$t_1y+x=at_1^2$, $t_2y+x=at_2^2$ and $t_3y+x=at_3^2$

The respective points of intersection are:

$(at_2.t_3,a(t_2+t_3))$, $(at_1.t_3,a(t_1+t_3))$ and $(at_1.t_2,a(t_1+t_2))$

Since these points lie on the hyperbola, Putting these points in $xy=c^2$:

$t_2.t_3(t_2+t_3)=\frac{c^2}{a^2}$ ...(1)
$t_3.t_1(t_3+t_1)=\frac{c^2}{a^2}$ ...(2)
$t_1.t_2(t_1+t_2)=\frac{c^2}{a^2}$ ...(1)

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I am not able to figure out how to proceed after this... Can someone please help me out?

Answer 1

I haven't checked the derivation of your three equations, but it is not difficult to see that they are equivalent to this system of only TWO equations: $$ _1+_2+_3=0 \\ _1_2_3=−{^2\over ^2}. $$ Hence there are infinitely many solutions: for instance you can choose $t_1$ at will and find $t_2$, $t_3$ from the above equations.