Question

In the figure $| z |= r$ is circumcircle of $\triangle ABC . D , E \,\&\, F$ are the middle points of the sides $BC , CA\, \&\, AB$ respectively, $AD$ produced to meet the circle at $L$. If $\angle CAD =\theta, AD = x , BD = y$ and altitude of $\triangle ABC$ from A meet the circle $|z|=r$ at $M, z_a, z_b \,\&\, z_c$ are affixes of vertices $A, B \,\&\, C$ respectively.
Affix of $M$ is -

• A. $2 z_b e^{ i 2 B }$
• B. $z_b e^{i(\pi-2 B)}$
• C. $z_b e^{i B}$
• D. $2 z_b e^{i B}$

Answer 1

Answer: (B)

Let affix of $M$ is $z_m$ and $\angle B O M=\pi-2 B$, then
$\frac{z_m-0}{z_b-0}=\frac{O M}{O B} e^{i(\pi-2 B)} $
$z_m=z_b e^{i(\pi-2 B)}$