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Question
Complex numbers z1,z2,z3 are the vertices A,B,C respectively, of an isosceles right-angled triangle with right angle at C. Then which of the following is true?
Complex numbers z1,z2,z3 are the vertices A,B,C respectively, of an isosceles right-angled triangle with right angle at C. Then which of the following is true?
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Solution
The correct option is B (z1−z2)2=2(z1−z3)(z3−z2).
Applying rotation about point B,
z1−z2z3−z2=√2eiπ/4⋯(1)
Applying rotation about point A,
z2−z1z3−z1=√2e−iπ/4⋯(2)
Multiplying (1) and (2) , we get
(z1−z2)(z2−z1)(z3−z2)(z3−z1)=2
(z1−z2)2=−2(z3−z2)(z3−z1)
=2(z1−z3)(z3−z2)
Applying rotation about point B,
z1−z2z3−z2=√2eiπ/4⋯(1)
Applying rotation about point A,
z2−z1z3−z1=√2e−iπ/4⋯(2)
Multiplying (1) and (2) , we get
(z1−z2)(z2−z1)(z3−z2)(z3−z1)=2
(z1−z2)2=−2(z3−z2)(z3−z1)
=2(z1−z3)(z3−z2)
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