Let `P(e^(itheta_1)), Q(e^(itheta_2)) and R(e^(itheta_3))` be the vertices of a triangle `PQR` in the Argand Plane. Theorthocenter of the triangle `PQR` is

Options

A. `e^(i(theta_(1)+theta_(2)+theta_(3))`
B. `2/3e^(i(theta_(1)+theta_(2)+theta_(3))`
C. `e^(i(theta_(1))+e^(itheta_(2))+e^(itheta_(3))`
D. none of these

Answer

C

Explanation

We have,
`|e^(itheta_(1))|=|e^(itheta_(2))|=|e^(itheta_(3))|=1`
`rArr OP=OQ=OR=1`, where O is the origin.
`rArr` Origin O is the circumference of `trianglePQR`.
The affix of the centroid is `1/3e^(itheta_(1))+e^(itheta_(2))+e^(itheta_(3))`.
Let z be the affix of the orthocenter. Since centroid divides the segment joining circumference and othocenter in the ratio `1:2`.
`therefore 1/3(e^(itheta_(1))+e^(itheta_(2))+e^(itheta_(3)))=(1 xx z + 2 xx 0)/(1+2) rArr z=e^(itheta_(1)) + e^(itheta_(2))+e^(itheta_(3))`