Question

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

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

Answer 1

Answer: (A), (B)

$Re\left(\frac{z}{z_0}\right)=1$
$z\overline{z_0}+z_0\overline{z}=2r^2$

Let $z=x+iy$
Hence
$|z|=r$
Represents $x^2+y^2=r^2$
Now $z_0=x_0+i{y}_0$ is a point on $x^2+y^2=r^2$.
Hence $|z_0|=r$ Since the center to the circle is at origin.
Using point contact form of tangent, we get the equation of tangent as
$x{x}_0+y{y}_0=r^2$
This is nothing but the real part of
$z.\overline{z_0}=r^2=|z_0{|}^2$
$Re\left(\frac{z}{z_0}\right)=1$
Now
Consider
$z\overline{z_0}+z_0\overline{z}=2r^2$
Upon simplifying, we get
$2x{x}_0+2y{y}_0=2r^2$
$x{x}_0+y{y}_0=r^2$
Hence equation of tangent.