Question

If $z=\frac{3}{2+cos\theta +isin\theta }$, then locus of $z$ is

• A. A straight line
• B. A circle having center on the y-axis
• C. A parabola
• D. A circle having center on the x-axis

Answer 1

The correct option is D A circle having center on the x-axis

Given, $z=\frac{3}{2+i\sin \theta +\cos \theta }$
$\therefore \cos \theta +i\sin \theta =\frac{3}{z}-2=\frac{3-2z}{z}$
Taking the modulus,
$\left|z\right|=|3-2z|$
Let $z=x+iy$
$\therefore x^2+y^2=9-12x+4x^2+4y^2\Rightarrow x^2+y^2-4x+3$
Circle with centre on x-axis