This was a problem in my recent Year 12 maths exam.
For which values of $\theta$ does the normal to point $P(a \cos \theta, b \sin \theta)$ on ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ intersects the hyperbola $xy=c^2$ at two points.
I haven't currently seen the answer key, but from I believe they likely made an error in solving the problem.
It can be easily proven that the $1$st and $3$rd quadrants both satisfy (excluding the asymptotes), and I suspect these would be the answers given. However, there are cases in the $2$nd and $4$th quadrant for which the conditions of the question are indeed satisfied.
These cases appear quite non-trivial and hard to manipulate, and I require assistance in trying to determine the conditions for $a,b,c$ for which $\theta$ does satisfy in the $2$nd and $4$th quadrant, and what is $\theta$ bounded by in these quadrants.