Question

If from a point P representing the complex number $z_1$ on the curve $|z|=2$, two tangents are drawn from P to the curve $|z|=1$, meeting at points $Q\left(z_2\right)$ and $R\left(z_3\right)$, then

• A. Complex number $\frac{(z_1+z_2+z_3)}{3}$ will be on the curve $|z|=1$
• B. $(\frac{4}{{\overline{z}}_1}+\frac{1}{{\overline{z}}_2}+\frac{1}{{\overline{z}}_3})(\frac{4}{z_1}+\frac{1}{z_2}+\frac{1}{z_3})=9$
• C. $arg\left(\frac{z_2}{z_3}\right)=\frac{2\pi }{3}$
• D. Orthocenter and circumcenter of $\Delta PQR$ will concide

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A Complex number $\frac{(z_1+z_2+z_3)}{3}$ will be on the curve $|z|=1$
B $(\frac{4}{{\overline{z}}_1}+\frac{1}{{\overline{z}}_2}+\frac{1}{{\overline{z}}_3})(\frac{4}{z_1}+\frac{1}{z_2}+\frac{1}{z_3})=9$
C $arg\left(\frac{z_2}{z_3}\right)=\frac{2\pi }{3}$
D Orthocenter and circumcenter of $\Delta PQR$ will concide

Since $OQ=1$ and $OP=2$, so $sin\left(\angle OPQ\right)=1/2$ and hence $\angle QPR=\pi /3$. Then $\angle PQR$ is equilateral. Also, $OM\perp QR$. Then from $\angle OMQ,OM=1/2$. Hence, $MN=1/2$. Then centroid of $\angle PQR$ lies on $|z|=1$.
AS PQR is an equilateral triangle, so orthocenter, circumcenter, and centroid will coincide. Now,
$\left|\frac{z_1+z_2+z_3}{3}\right|=1$
or $|z_1+z_2+z{|}^2=9$
or $(z_1+z_2+z_3)({\overline{z}}_1+{\overline{z}}_2+{\overline{z}}_3)=9$
or $(\frac{4}{{\overline{z}}_1}+\frac{1}{{\overline{z}}_2}+\frac{1}{{\overline{z}}_3})(\frac{4}{z_1}+\frac{1}{z_2}+\frac{1}{z_3})=9$ and $\angle QOR={120}^o$