1.

LetA (z_1 )andB(z_2 )be lying onthe curve|z-3 - 4i| = 5, where|z_1 |is maximum. Now,A(z_1) is rotated about the origin inanticlockwise direction through 90^(@)reachingatP(z_0). IfA, B and Pare collinear thenthevalue of(|z_0 - z_ 1 | * |z_ 0- z_ 2 |)is _______.

Answer»


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

This circlepasses thorughtthe origin.
`z_(1)` lies on circlesuch that `|z_(1)|` is MAXIMUM.
Therefore ,`z_(1) ` is at other end of thediameter through origin.
`therefore z_(1) = 6+ 8i`
OAis 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`


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