Question

If $z=4\cos \theta (\cos \theta +i\sin \theta ),\theta \in R$, then the locus of $z$ on the argand plane is

• A. an ellipse with centre $(0,2)$
• B. a circle with centre $(0,2)$
• C. an ellipse with centre $(2,0)$
• D. a circle with centre $(2,0)$

Answer 1

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

$z=4\cos \theta (\cos \theta +i\sin \theta )\Rightarrow z=4{\cos }^2\theta +i\left(4\sin \theta \cos \theta \right)\text{Let}z=x+iy\Rightarrow x=4{\cos }^2\theta =2(1+\cos 2\theta )$
and $y=2\cdot 2\sin \theta \cos \theta =2\sin 2\theta$
$\Rightarrow (x-2{)}^2+y^2=4$