Suppose a smooth, connected curve $C$ in $R^2$ is orthogonal to all hyperbolae $xy = a$ whenever they coincide. I'd like to find the point(s) of intersection of $C$ with the hyperbola $xy = 16$ given that $C$ contains the point (1,1).
I figure that I should write the curve $C$ as some general parametrization $C(t) = (p(t),q(t))$ and use the fact that, if $C(t)$ intersects the hyperbola $H_a(t) = (t,a/t)$ at $t = t_0$, then $$C'(t_0) \cdot H'_a(t_0) = tp(t_0) + (a/t_0)q(t_0) = 0.$$
Also, since both $C(t)$ and $H_1(t)$ contain the point (1,1), this gives me some specific information about where $C(t)$ is orthogonal to the particular hyperbola $H_1(t)$.
This question is from a practice exam and does not seem very advanced, but I keep getting stuck. If someone could give me a nudge in the right direction, I'd be grateful. Thanks.