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Question

Let A(z1) and B(z2) be two points lying on the curve z34i=25¯¯¯z3+4i where |z1| is maximum. Now, A(z1) is rotated about the origin in anti-clockwise direction through 90 reaching at P(z0). If A,B and P are collinear, then the value of |z0z1||z0z2| is

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Solution

z34i=25¯¯¯z3+4i
(z(3+4i))¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(z(3+4i))=25|z(3+4i)|=5 (z¯¯¯z=|z|2)
This represents a circle with centre at (3,4) and radius 5.
Since, |z1| is maximum, A occurs along the diameter.
z1=6+8i


Now, z0=(6+8i)i=8+6i
PAPB=OP2|z0z1||z0z2|=100

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