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.

  • 3
    $\begingroup$ How are points A and B defined ? $\endgroup$ – StephenG Aug 18 '17 at 1:24
  • $\begingroup$ They are the intersections of the normal with the hyperbola. @StephenG $\endgroup$ – Sharky Kesa Aug 18 '17 at 1:54
  • $\begingroup$ Did u do any working algebraically? $\endgroup$ – bigfocalchord Aug 18 '17 at 1:56
  • $\begingroup$ @dydxx Yeah, I did. Using discriminant, I can get it down to $(a^2-b^2)^2\sin(2\theta)+8abc^2<0$, assuming $\theta$ is in 2nd or 4th quadrant. This expression is quite weird to deal with. I've tried using bounding arguments without success. $\endgroup$ – Sharky Kesa Aug 18 '17 at 1:58
  • $\begingroup$ @SharkyKesa Yeah I got to that part as well, the inequality is quite pesky $\endgroup$ – bigfocalchord Aug 18 '17 at 2:01

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