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 $L$ is -

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

Answer 1

Answer: (A)

Let affix of $L$ is $z_L$ and $\angle B O L=2(A-\theta)$, then
$\frac{z_L-0}{z_b-0}=e^{i(2 A-2 \theta)} $
$z_L=z_b e^{i(2 A-2 \theta)}$