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Question

If P and Q are respected by the complex numbers z1 and z2 such that |1z1+1z2|=|1z11z2| then the circumcentre of ΔOPQ (where O is the origin ) is

A
z1z22
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B
z1+z22
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C
z1+z23
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D
z1+z2
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Solution

The correct option is D z1+z22
1z1+1z2=1z11z2
|z2+z1|=|z2z1|
|z2+z1|2=|z2z1|2
|z2+z1|2+|z2+z1|2=2|z1|2+2|z2|2
2|z2z1|2=2|z1|2+2|z2|2
|z2z1|2=|z1|2+|z2|2
Thus OPQ is rightangled at O.
Hence mid-point of hypotenuse is the circumcentre, i.e., z1+z22.
Hence the circumcentre of OPQ is z1+z22.

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