Question

If $|z|=2$ and locus of 5z -1 is the circle having radius a and $z_1^2+z_2^{2}2z_1{z}_{2}cos\theta =0$, then $|z_1|:|z_2|=$

• A. a : 1
• B. 2a : 1
• C. a : 10
• D. none o these

Answer 1

Answer: (C)

a : 10

Given $|z|=2$
Let $\alpha =5z-1\Rightarrow |\alpha +1|=5|z|=5\left(2\right)=10$
$\Rightarrow |a+1|=10$

$\therefore$ locus of $\alpha$ i.e., $5z-1$ is the circle having centre at $-1$ and radius $10$.

$\therefore a=10$.

Again $z_1^2+z_2^2-2z_1{z}_2\cos \theta =0$

$\Rightarrow {\left(\frac{z_1}{z_2}\right)}^2-2\frac{z_1}{z_2}\cos \theta +1=0$

$\Rightarrow \frac{z_1}{z_2}=\frac{2\cos \theta \pm \sqrt{4{\cos }^2\theta -4}}{2}=\cos \theta \pm i\sin \theta$

$\Rightarrow \left|\frac{z_1}{z_2}\right|=|\cos \theta \pm i\sin \theta |=1=\frac{a}{10}$

$\therefore |z_1|:|z_2|=a:10$