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