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Question: If amp. [z<sub>1</sub> (z<sub>3</sub> – z<sub>2</sub>)] = amp. [z<sub>3</sub> (z<sub>2</sub> – z<sub...

If amp. [z1 (z3 – z2)] = amp. [z3 (z2 – z1)], then –

A

z1, z2, z3 are collinear

B

z1, z2, z3 are the vertices of an equilateral D

C

O, z1, z2, z3 are concyclic

D

None of these

Answer

O, z1, z2, z3 are concyclic

Explanation

Solution

Sol. Amp. [z1 (z3 – z2)] = Amp. [z3 (z2 – z1)]

\ amp. [z1(z3−z2)][z3(z2−z1)]\frac{\lbrack z_{1}(z_{3} - z_{2})\rbrack}{\lbrack z_{3}(z_{2} - z_{1})\rbrack} = 0

z1(z3−z2)z3(z2−z1)\frac{z_{1}(z_{3} - z_{2})}{z_{3}(z_{2} - z_{1})} is purely real

z1 , z2 , z3 are concyclic

For four concyclic points (z1−z3)(z2−z4)(z1−z4)(z2−z3)\frac{(z_{1} - z_{3})(z_{2} - z_{4})}{(z_{1} - z_{4})(z_{2} - z_{3})} is purely real.