Let ` A (z_1 ) and B(z_2 )` be lying on the curve `|z-3 - 4i| = 5`, where `|z_1 |` is maximum. Now, `A(z_1)` is rotated about the origin in anticlockwise direction through `90^(@)` reaching at `P(z_0)`. If `A, B and P ` are collinear then the value of ` (|z_0 - z_ 1 | * |z_ 0 - z_ 2 |)` is _______.

Answer

100

Explanation

We have `|z-(3+4i)| = 5`
So, z lies on the circle having centre at `3+4i` and radius 5.

This circle passes thorught the origin.
`z_(1)` lies on circle such that `|z_(1)|` is maximum.
Therefore ,`z_(1) ` is at other end of the diameter through origin.
`therefore z_(1) = 6+ 8i`
OA is rotated by `90^(@)` in anticlcokwise direction.
So, `z_(0) = i(6+8i) = - 8+ 6i`
Since A,B and P are colliner.
`|z_(0)-z_(1)|.|z_(0)-z_(2)| = Pa.PB =OP^(2) = 10^(2) = 100`