Question

If \(z\) represents point on circle \(|z|=2\) then locus of \(z+\frac1z\) is

• A. parabola
• B. circle
• C. ellipse
• D. hyperbola

Hint:

Parametrize complex circle with exponential form.

Answer 1

The Correct Option is C

Step 1: Let \(z=2e^{i\phi}\).
Step 2: \[ w=z+\frac1z=2e^{i\phi}+\frac12 e^{-i\phi}. \]
Step 3: Separate real and imaginary parts: \[ x=2\cos\phi+\frac12\cos\phi=\frac52\cos\phi, \] \[ y=2\sin\phi-\frac12\sin\phi=\frac32\sin\phi. \]
Step 4: Eliminate \(\phi\): \[ \frac{x^2}{(5/2)^2}+\frac{y^2}{(3/2)^2}=1. \] This is ellipse. Hence → (C).