Question

If from a variable point P representing the complex number $z_1$ on the curve $|z|=4$, two tangents are drawn to the curve $|z|=2$, meeting it at points $Q\left(z_2\right)$ and $R\left(z_3\right)$, then which of the following statement(s) is(are) correct?

• A. Triangle PQR is isosceles.
• B. The locus of centroid of triangle $PQR$ is $|z|=1$.
• C. The circumradius of triangle $PQR$ is 2 .
• D. The radius of circle inscribed in triangle PQR is 1 .

Answer 1

Answer: (C), (D)

$\therefore$ From above figure, $\cos (\angle POR)=\frac{OR}{OP}\frac{2}{4}=\frac{1}{2}$
$\Rightarrow \angle POR=\frac{\pi }{3}=\angle POQ\Rightarrow \angle OPR=\angle OPQ=30^{\circ }$
$\Rightarrow \angle QPR=60^{\circ }$....(1)
Also, in $△PQR,PQ=PR$....(2)
$\therefore$ From (1) and (2), we get $△PQR$ is equilateral $\Rightarrow (A)$ is incorrect.
Also, $PQOR$ are concyclic and $\angle OQP$ and $\angle ORP=90^{\circ }$ So, circumcentre of $△PQR$ passes through $O(0,0)$ and $OP$ is diameter of it.
So, circumcentre of $△PQR=$ mid point of $OP$
$=\left(\frac{0+4\cos \theta }{2},\frac{0+4\sin \theta }{2}\right)=(2\cos \theta ,2\sin \theta )$
$=\text{ centroid of }△PQR[As,\Delta PQR\text{ is equilateral. }]$
$\therefore$ The locus of centroid of $△PQR$ is $|z|=2\Rightarrow$ (B) is incorrect. Also, circumradius of $△PQR=\frac{OP}{2}=\frac{4}{2}=2\Rightarrow (C)$ is correct. As, $r=\frac{R}{2}=\frac{2}{2}=1$ (As, $△PQR$ is equilateral.) $\Rightarrow$ radius of circle inscribed in $△PQR$ is $1.\Rightarrow$ (D) is correct.