Question

If t and c are two complex numbers such that $\left|t\right|\ne \left|c\right|,\left|t\right|=1$ and z = (at + b) /(t - c), z = x + iy. Locus of z is
(where a, b are complex numbers)

• A. line segment
• B. straight line
• C. circle
• D. None of these

Answer 1

The correct option is C circle

$z=\frac{at+b}{t-c}\Rightarrow t=\frac{b+cz}{z-a}$
Now, $\left|t\right|=1$
$\Rightarrow \left|\frac{b+cz}{z-a}\right|=1$
$\Rightarrow \left|\frac{z+\frac{b}{c}}{z-a}\right|=\frac{1}{\left|c\right|}(\ne 1$ as $\left|c\right|\ne \left|t\right|)$
$\Rightarrow$ locus of z is a circle