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Question: If complex numbers \(z_{1},z_{2}\) and \(z_{3}\) represent the vertices A, B and C respectively of a...

If complex numbers z1,z2z_{1},z_{2} and z3z_{3} represent the vertices A, B and C respectively of an isosceles triangle ABC of which ∠C\angle C is right angle, then correct statement is

A

z12+z22+z32=z1z2z3z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = z_{1}z_{2}z_{3}

B

(z3−z1)2=z3−z2(z_{3} - z_{1})^{2} = z_{3} - z_{2}

C

(z1−z2)2=(z1−z3)(z3−z2)(z_{1} - z_{2})^{2} = (z_{1} - z_{3})(z_{3} - z_{2})

D

(z1−z2)2=2(z1−z3)(z3−z2)(z_{1} - z_{2})^{2} = 2(z_{1} - z_{3})(z_{3} - z_{2})

Answer

(z1−z2)2=2(z1−z3)(z3−z2)(z_{1} - z_{2})^{2} = 2(z_{1} - z_{3})(z_{3} - z_{2})

Explanation

Solution

Sol. BC=ACBC = AC and ∠C=π/2\angle C = \pi/2

By rotation about C in anticlockwise sense CB=CAeiπ/2CB = CAe^{i\pi/2}

⇒ (z2−z3)=(z1−z3)eiπ/2=i(z1−z3)(z_{2} - z_{3}) = (z_{1} - z_{3})e^{i\pi/2} = i(z_{1} - z_{3})

⇒ (z2−z3)2=−(z1−z3)2(z_{2} - z_{3})^{2} = - (z_{1} - z_{3})^{2}⇒ z22+z32−2z2z3=−z12−z32+2z1z3z_{2}^{2} + z_{3}^{2} - 2z_{2}z_{3} = - z_{1}^{2} - z_{3}^{2} + 2z_{1}z_{3}

⇒ z12+z22−2z1z2=2z1z3+2z2z3−2z32−2z1z2z_{1}^{2} + z_{2}^{2} - 2z_{1}z_{2} = 2z_{1}z_{3} + 2z_{2}z_{3} - 2z_{3}^{2} - 2z_{1}z_{2}

⇒ (z1−z2)2=2[(z1z3−z32)−(z1z2−z2z3)](z_{1} - z_{2})^{2} = 2\lbrack(z_{1}z_{3} - z_{3}^{2}) - (z_{1}z_{2} - z_{2}z_{3})\rbrack

⇒ (z1−z2)2=2(z1−z3)(z3−z2).(z_{1} - z_{2})^{2} = 2(z_{1} - z_{3})(z_{3} - z_{2}).