Question

If $z=4\cos \theta (\cos \theta +i\sin \theta )$, then locus of $z$ is

• A. circle with centre $(-4,0)$
• B. circle with centre $(2,0)$
• C. ellipse
• D. hyperbola

Answer 1

The correct option is B circle with centre $(2,0)$

Let, $z=x+iy$

$\Rightarrow x+iy=4{\cos }^2\theta +2\sin 2\theta i$

$\Rightarrow x=2(1+\cos 2\theta ),y=2\sin 2\theta$

$\Rightarrow x-2=2\cos 2\theta ,y=2\sin 2\theta$

$\frac{x-2}{2}=\cos 2\theta ,\frac{y}{2}=\sin 2\theta$

squaring and adding we get,

${\cos }^{2}2\theta +si{n}^{2}2\theta =1$

${(x-2)}^2+y^2=4$. which is circle centred at $(2,0)$