The maximum area of the triangle formed by the complex coordinates `z, z_1,z_2` which satisfy the relations `|z-z_1|=|z-z_2|` and `|z-(z_1+z_2 )/2|<= r`,where `r>|z_1-z_2|` is

Options

A. `1/2|z_(1)-z_(2)|^(2)`
B. `1/2|z_(1)-z_(2)|r`
C. `1/2|z_(1)-z_(2)|^(2)r^(2)`
D. `1/2|z_(1)-z_(2)|r^(2)`

Answer

B

Explanation

Let A(z), `B(z_(1))` and `C(z_(2))` be the points forming triangle ABC.
It is given that,
`|z-z_(1)|=|z-z_(2)|`
`rArr` AB=AC
`rArr` A lies on or the perpendicular bisector of BC.

Also,
`|z-(z_(1)+z_(2))/(2)| le r`
`rArr` A(z) lies on or interior of the circle having center at the mid-points of BC and radius r`( gt BC)`
Now, Area of `triangle ABC`.
`1/2 (BC xx AD)=1/2|z_(1)-z_(2)||z-(z_(1)+z_(2))/(2)| le 1/2|z_(1)-z_(2)|r`.
Hence, the maximum area of `triangleABC` is `1/2|z_(1)-z_(2)|r`.