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Isosceles Right Triangle Vertices in Complex Numbers | JEE

JEE Maths question with a full step-by-step solution.

Question
If z1,z2,z3z_1, z_2, z_3 are the vertices of an isosceles right-angled triangle, right-angled at the vertex z2z_2, then (z1−z2)2+(z3−z2)2(z_1 - z_2)^2 + (z_3 - z_2)^2 equals
A00correct
B(z1−z3)2(z_1 - z_3)^2
C(z1+z32)2\left(\dfrac{z_1 + z_3}{2}\right)^2
DNone
Solution
Step 1: A right angle at z2z_2 with equal legs means z3−z2z_3 - z_2 is obtained from z1−z2z_1 - z_2 by a 90°90° rotation:
z3−z2=±i(z1−z2)z_3 - z_2 = \pm i(z_1 - z_2)
 Step 2: Therefore (z3−z2)2=−(z1−z2)2(z_3 - z_2)^2 = -(z_1 - z_2)^2, giving
(z1−z2)2+(z3−z2)2=0(z_1 - z_2)^2 + (z_3 - z_2)^2 = 0
 Correct answer: (1)
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