Question

The complex number associated with the centre of the part of a circle represented by arg $\left(\frac{z-3i}{z-2i+4}\right)=\frac{\pi }{4}$ is

• A. $\frac{1}{2(5i+5)}$
  
• B. $\frac{1}{2}(5i-5)$
  
• C. $\frac{1}{2}(9i+5)$
  
• D. $\frac{1}{2}(9i-5)$

Answer 1

The correct option is D $\frac{1}{2}(9i-5)$

The given equation implies that the points representing the complex number 3i and -4 +2i subtend an angle $\frac{\pi }{4}$ at the circumference of the circle. As such these points subtend an angle $\frac{\pi }{2}$ at the centre of the circle. If $z_0$ is the corresponding complex number associated with the centre then $\frac{z_0-3i}{z_0-(-4+2i)}=e^{i\pi /2}=i\Rightarrow z_0=\frac{1}{2}(9i-5)$