While studying coordinate geometry I came across a question -
Prove that chords of a hyperbola, which touch the conjugate hyperbola, are bisected at the point of contact.
Since everyone hyperbola can be transformed into a standard hyperbola, whose transverse and conjugate axis are along the coordinate axis by suitable shifting of origin, I can prove this question for a standard hyperbola.
Unfortunately, in spite of many tries, I'm not able to prove this statement. Any help will be greatly appreciated.
Here's my try, it includes the figure (in case if statement is not clear, also note that the figure is not to scale.)
It seems I've proved that the point can't bisect the chord, which sounds contradictory. I guess something is wrong in my proof.