InterviewSolution
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InterviewSolution
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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. |
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Answer» `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. |
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