Question

Equation of tangent drawn to circle $|z|=r$ at the point $A\left(z_0\right)$ is

• A. $\text{Re}\left(\frac{z}{z_0}\right)=1$
• B. $z{\overline{z}}_0-z_0\overline{z}=2r^2$
• C. $\text{Im}\left(\frac{z}{z_0}\right)=1$
• D. $z{\overline{z}}_0+z_0\overline{z}=2r^2$

Answer 1

Answer: (A), (D)

$z{\overline{z}}_0+z_0\overline{z}=2r^2$

$\angle OAP=\frac{\pi }{2}$
$\Rightarrow \frac{z-z_0}{z_0}$ is purely imaginary
$\Rightarrow \frac{z-z_0}{z_0}+\frac{\overline{z}-{\overline{z}}_0}{{\overline{z}}_0}=0$
$\Rightarrow \frac{z}{z_0}+\frac{\overline{z}}{{\overline{z}}_0}=2\cdots \left(1\right)$
$\Rightarrow \text{Re}\left(\frac{z}{z_0}\right)=1$

From $\left(1\right)$,
$z{\overline{z}}_0+z_0\overline{z}=2|z_0{|}^2=2r^2$