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If |(z-z(1))//(z-z(2))| = 3, where z(1) ...

If `|(z-z_(1))//(z-z_(2))| = 3`, where `z_(1)` and `z_(2)` are fixed complex numbers and z is a variable complex complex number, then z lies on a

A

circle with `z_(1)` as its interior point

B

circle with `z_(2)` as its interior point

C

circle with `z_(1)` as its exterior point

D

circle with `z_(2)` as its exterior point

Text Solution

Verified by Experts

The correct Answer is:
B, C

Let internal bisectors of `/_APB` meet the line
joining A and B at `P_(1)` and `P_(2)` respectively . Hence,
`AP_(1):P_(1)B -= PA:PB -= 3:1` (internal division )
`AP_(2):P_(2)B -=PA:PB -= 3:1` (external division )
Thus, `P_(1)` and `P_(2)` are fixed points . Also
`/_P_(1) PP_(2) = (pi)/(2)`
Thus, P lies on a circle having `P_(1)P_(2)` as it diametet. Claerly,`B(z_(2))` lies inside this cricle.
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