I have a question that says the following:
A line thought the origin meets the circle $x^2+y^2=a^2$ at P and the hyperbola $x^2-y^2=a^2$ at Q. What is the locus of the point of intersection of the tangent at P to the circle with the tangent at Q to the hyperbola?
So, the few things I noticed right away were, clearly the given circle is the auxiliary circle to the hyperbola. So, what I did was, I tried to take the points P and Q in parametric form and write equation of tangents to the circle and hyperbola using that, but I hit a snag:
Can I use the same parametric angle for both P and Q, i.e, "$\theta$" ??
Or do I need to use two different parametric angles for P and Q, in which case, I think I would be better off assuming the equation of the line through the origin as $y = mx$ and then finding P and Q by solving this equation of the line and the equations of the curves simultaneously and then find the tangents at those points? Is this approach correct?