Question

If $\left|z\right|=2$ and locus of 5z - 1 is a circle having radius `a' and $z_1^2+z_2^2-2z_1{z}_2\cos \theta =0$ then $\left|z_1\right|:\left|z_2\right|$ is equal to

• A. a
• B.
• C.
• D. None

Answer 1

The correct option is C

Let $\omega =5z-1⟹|\omega +1|=5\left|z\right|=10$
$\therefore$locus is a circle centre at (-1, 0) and radius 10
$\therefore a=10$
Again $z_1^2+z_2^2-{2z}_1{z}_2\cos \theta =0$
$⟹{\left(\frac{z_1}{z_2}\right)}^2-2\left(\frac{z_1}{z_2}\right)cos\theta +1=0$
$⟹\frac{z_1}{z_2}=cos\theta \pm isin\theta ⟹\left|\frac{z_1}{z_2}\right|=1=\frac{a}{10}$