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'Implicitly' printed from https://nrich.maths.org/
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?