Question

If $t$ and $c$ are two complex numbers such that $|t|\ne |c|,|t|=1$ and $z=\frac{at+b}{t-c},z=x+iy$, then locus of $z$ is (where $a,b$ are complex numbers)

• A. line segment
• B. straight line
• C. circle
• D. ellipse

Answer 1

The correct option is C circle

$z=\frac{at+b}{t-c}\Rightarrow t=\frac{b+cz}{z-a}$
Now,
$|t|=1$
$\Rightarrow \left|\frac{b+cz}{z-a}\right|=1$
$\Rightarrow \frac{|z+\frac{b}{c}|}{|z-a|}=\frac{1}{|c|}(\ne 1\text{as}|c|\ne |t|)$
$\Rightarrow$ locus of $z$ is a circle, since we know that $\left|\frac{z-z_1}{z-z_2}\right|=k$ denotes a circle if $k\ne 1$