Question

Consider the circle $|z-5i|=3$ and two points $z_1$ and $z_2$ on it such that $|z_1|<|z_2|$ and $\arg \left(z_1\right)=\arg \left(z_2\right)=\frac{\pi }{3}.$ A tangent is drawn at $z_2$ to the circle, which cuts the real axis at $z_3,$ then $|z_3|$ is

• A. $3$
• B. $\sqrt{13}$
• C. $\sqrt{11}$
• D. $4$

Answer 1

The correct option is C $\sqrt{11}$


$O,A,B,D$ are concyclic points.
$\cos 60°=\frac{BD}{AD}=\frac{3}{AD}$

$AD=6,OD=5$
$|z_3|=OA=\sqrt{36-25}=\sqrt{11}$