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Question: If z<sub>1</sub>, z<sub>2</sub>, z<sub>3</sub> are three distinct complex numbers and a, b, c are th...

If z1, z2, z3 are three distinct complex numbers and a, b, c are three positive real numbers such that a∣z2−z3∣\frac{a}{|z_{2} - z_{3}|}=b∣z3−z1∣\frac{b}{|z_{3} - z_{1}|}=c∣z1−z2∣\frac{c}{|z_{1} - z_{2}|}, then a2(z2−z3)\frac{a^{2}}{(z_{2} - z_{3})}+b2(z3−z1)\frac{b^{2}}{(z_{3} - z_{1})}+c2(z1−z2)\frac{c^{2}}{(z_{1} - z_{2})}=

A

0

B

Abc

C

3abc

D

A + b + c

Answer

0

Explanation

Solution

Sol. Let a∣z2−z3∣\frac{a}{|z_{2} - z_{3}|}= b∣z3−z1∣\frac{b}{|z_{3} - z_{1}|}= c∣z1−z2∣\frac{c}{|z_{1} - z_{2}|}= l (say)

Ž a = |z2 – z3|, b = l |z3 – z1|, c = l |z1 – z2|

Ž a2 = l2 |z2 – z3|2 = l2 (z2 – z3)(zˉ2−zˉ3)({\bar{z}}_{2} - {\bar{z}}_{3})

\ a2(z2−z3)\frac{a^{2}}{(z_{2} - z_{3})} = l2(zˉ2−zˉ3)({\bar{z}}_{2} - {\bar{z}}_{3})

Similarly, a3(z3−z1)\frac{a^{3}}{(z_{3} - z_{1})}= l2(zˉ3−zˉ1)({\bar{z}}_{3} - {\bar{z}}_{1}) andc2(z1−z2)\frac{c^{2}}{(z_{1} - z_{2})}

= l2(zˉ1−zˉ2)({\bar{z}}_{1} - {\bar{z}}_{2})

\ a2z2−z3\frac{a^{2}}{z_{2} - z_{3}}+b2z3−z1\frac{b^{2}}{z_{3} - z_{1}}+|c2z1−z2\frac{c^{2}}{z_{1} - z_{2}}= 0.