If z has real part x and imaginary part y, z = |z|(cos(θ) + sin(θ)i), where $-\pi\small <$ θ ${\small\leqslant}\, \pi$,

and (unless z = 0) 1/z = (1/|z|)(cos(θ) - sin(θ)i).

It's easy to verify that the product of those expressions is 1 (but you can assume their validity anyway).

The amount of detail that you give in your answer depends on what you've already learnt, but you should obtain the real and imaginary parts of z + 1/z by using the equations I've given above.

Thus x and y can be thought of as coordinates for a point on an x-y plane (known as an Argand diagram). You can assume that |z| = 2 means the point P(x, y) lies on a circle of radius 2 and equation x² + y² = 2².

Let Q be a point on the same Argand diagram representing the complex number z + 1/z.

You can verify algebraically that the corresponding equation for the locus of Q is x²/25 + y²/9 = (1/2)² = 1/4, where x and y are now the real and imaginary parts of z + 1/z.

You can assume that is the equation of an ellipse.