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### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### Advanced mathematics

### For younger learners

# 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

### Key questions

### Possible extension

### Possible support

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Age 16 to 18

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

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.

Are you clear which parts are variables and which parts are
constants?

How can we find $X$ in terms of $r$ directly?

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 $X$?

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