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Question

# If z1,z2,z3 non-zero, non-collinear complex numbers such that 2z1=1z2+1z3, then the points z1,z2,z3 lie

A
in the interior of a circle
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B
on a circle passing through origin
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C
in the exterior of a circle
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D
None of these
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Solution

## The correct option is B on a circle passing through originWe have, 2z1=1z2+1z3=z3+z2z2z3 ⇒z1=2z2z3z2+z3 Now, (z2−z4z1−z4)(z1−z3z2−z3) =⎛⎜ ⎜ ⎜⎝z2−z42z2z3z2+z3−z4⎞⎟ ⎟ ⎟⎠⎛⎜ ⎜ ⎜⎝2z2z3z2+z3−z3z2−z3⎞⎟ ⎟ ⎟⎠=z22z2z3z2+z3(z3(z2−z3)(z2−z3)(z2−z3)) [taking z4=0] =12 (a real number). Hence, points z1,z2,z3 and origin are concyclic and therefore, z1,z2,z3 lie on a circle passing through the origin.

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