Question

If `theta` is real and `z_1, z_2` are connected by `z1 2+z2 2+2z_1z_2costheta=0,` then prove that the triangle formed by vertices `O ,z_1a n dz_2` is isosceles.

Answer 1

`z_(1)^(2) + z_(2)^(2) + 2z_(1)z_(2) cos theta = 0`
`or ((z_(1))/(z_(2)))^(2) + 2(z_(1)/(z_(2))) cos theta + 1 =0`
`or ((z_(1))/(z_(2)) + cos theta )^(2) = - (1-cos^(2) theta) = - sin^(2) theta`
` or (z_(1))/(z_(2)) = - cos theta pm i sin theta`
`or |(z_(1))/(z_(2))| = sqrt((-costheta)^(2) + sin^(2) theta)=1`
`or |z_(1)|=|z_(2)|`
`or |z_(1) -0|=|z_(2)-0|`
Thus, triangle with vertices O,`z_(1),z_(2)` is iscosceles.