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

Answer 1

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

Orthocentre and circumcenter of $\Delta PQR$ will coincide



Here $OQ=1$ and $OP=2$, so $\sin \left(\angle OPQ\right)=\frac{1}{2}$

$\Rightarrow \angle OPQ=\frac{\pi }{6}\Rightarrow \angle QPR=\frac{\pi }{3}$.

$PQ=PR$. Then $\angle PQR$ is equilateral.

Also, $OM\perp QR$. $\angle OQM=\frac{\pi }{6}$,

$\sin \left(\angle OQM\right)=\frac{OM}{OQ}\Rightarrow OM=\frac{1}{2}$

$\Rightarrow MN=\frac{1}{2}$

$\Rightarrow$ Centroid of $\Delta PQR$ lies on $|z|=1$

As $PQR$ is an equilateral triangle, orthocentre,
circumcentre and centroid will coincide.

$\Rightarrow \left|\frac{z_1+z_2+z_3}{3}\right|=1$

$\Rightarrow |z_1+z_2+z_3{|}^2=9$

$\Rightarrow (z_1+z_2+z_3)(\overline{z_1}+\overline{z_2}+\overline{z_3})=9$

$\Rightarrow (\frac{4}{\overline{z_1}}+\frac{1}{\overline{z_2}}+\frac{1}{\overline{z_3}})(\frac{4}{\overline{z_1}}+\frac{1}{\overline{z_2}}+\frac{1}{\overline{z_3}})=9$

$[\because |z_1|=2\Rightarrow z_1\overline{z_1}=4|z_2|=|z_3|=1\Rightarrow z_2\overline{z_2}=z_3\overline{z_3}=1)]$

$arg\left(\frac{z_2}{z_3}\right)=arg{z}_2-arg{z}_3=\angle QOR$

$=\angle QOM+\angle MOR=\frac{\pi }{3}+\frac{\pi }{3}=\frac{2\pi }{3}$

Hence, option $\left(A\right),\left(B\right),\left(C\right)$ and $\left(D\right)$ all are correct.