Z1,Z2,Z3 are complex numbers and p, q, r ,are real numbers such that pz2−z3=qz3−z1=rz1−z2.Then p2z2−z3+q2z3−z1+r2z1−z2=

We have p2z2−z32=q2z3−z12=r2z1−z22=k (say)⇒p2z2−z3z¯2−z¯3=q2z3−z1z¯3−z¯1=r2z1−z2z¯1−z¯2=k⇒p2z2−z3=kz¯2−z¯3,q2z3−z1=kz¯3−z¯1,r2z1−z2=kz¯1−z¯2⇒p2z2−z3+q2z3−z1+r2z1−z2=kz¯2−z¯3+z¯3−z¯1+z¯1−z¯2=0.

Z1,Z2,Z3 are complex numbers and p, q, r ,are real numbers such that pz2−z3=qz3−z1=rz1−z2.Then p2z2−z3+q2z3−z1+r2z1−z2=

We have p2z2−z32=q2z3−z12=r2z1−z22=k (say)⇒p2z2−z3z¯2−z¯3=q2z3−z1z¯3−z¯1=r2z1−z2z¯1−z¯2=k⇒p2z2−z3=kz¯2−z¯3,q2z3−z1=kz¯3−z¯1,r2z1−z2=kz¯1−z¯2⇒p2z2−z3+q2z3−z1+r2z1−z2=kz¯2−z¯3+z¯3−z¯1+z¯1−z¯2=0.

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$Z_{1}, Z_{2}, Z_{3}$ are complex numbers and p, q, r ,are real numbers such that $\frac{p}{|z_{2} - z_{3}|} = \frac{q}{|z_{3} - z_{1}|} = \frac{r}{|z_{1} - z_{2}|} .$ Then $\frac{p^{2}}{z_{2} - z_{3}} + \frac{q^{2}}{z_{3} - z_{1}} + \frac{r^{2}}{z_{1} - z_{2}} =$

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Detailed Solution

We have $\frac{p^{2}}{{|z_{2} - z_{3}|}^{2}} = \frac{q^{2}}{{|z_{3} - z_{1}|}^{2}} = \frac{r^{2}}{{|z_{1} - z_{2}|}^{2}} = k$ (say)
$\begin{aligned} \Rightarrow \frac{p^{2}}{(z_{2} - z_{3}) ({\bar{z}}_{2} - {\bar{z}}_{3})} = \frac{q^{2}}{(z_{3} - z_{1}) ({\bar{z}}_{3} - {\bar{z}}_{1})} = \frac{r^{2}}{(z_{1} - z_{2}) ({\bar{z}}_{1} - {\bar{z}}_{2})} = k \\ \Rightarrow \frac{p^{2}}{z_{2} - z_{3}} = k ({\bar{z}}_{2} - {\bar{z}}_{3}), \frac{q^{2}}{z_{3} - z_{1}} = k ({\bar{z}}_{3} - {\bar{z}}_{1}), \frac{r^{2}}{z_{1} - z_{2}} = k ({\bar{z}}_{1} - {\bar{z}}_{2}) \\ \Rightarrow \frac{p^{2}}{z_{2} - z_{3}} + \frac{q^{2}}{z_{3} - z_{1}} + \frac{r^{2}}{z_{1} - z_{2}} = k ({\bar{z}}_{2} - {\bar{z}}_{3} + {\bar{z}}_{3} - {\bar{z}}_{1} + {\bar{z}}_{1} - {\bar{z}}_{2}) = 0 . \end{aligned}$