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Implicitly

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?