### Why do this problem?

This problem is a way to get into implicit functions using the familiar mathematics of quadratic equations. It would be a good discussion focus for the introduction of the topic of implicit functions. The third part allows students to make use of their calculus.

### Possible approach

The first part of this question would make a good starter, the second and third parts would be well suited for individual calculation; they could be approached experimentally/numerically or using algebra.

### Key questions

Are you clear which parts are variables and which parts are constants?
How can we find $X$ in terms of $r$ directly?

### Possible extension

For an exercise in complex numbers, you might try these extensions:

Which purely imaginary values of $r$ give purely imaginary or purely real values of $X$?
Which complex values of $r$ give real values of $X$?

### Possible support

Start off searching numerically for more real values of $r$ which give a real value of $X$. Who can find the smallest such value?