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P is a point on the argand diagram on th...

P is a point on the argand diagram on the circle with OP as diameter two points taken such that `angle POQ = angle QOR = theta`. If O is the origin and P, Q, R are are represented by complex `z_1, z_2, z_3` respectively then show that `z_2^2cos2 theta =z_1z_3 cos^2theta`

Text Solution

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`z_(2)=(OQ)/(OP)z_(1)e^(itheta) = cos thetaz(1)e^(itheta)" "(1)`
`z_(3) = (OR)/(OP)z_(1)e^(12theta) = cos 2thetaz_(1)e^(i2theta)" "(2)`
Form (1),`z_(2)^(2) = cos^(2) theta z_(1)^(2) e^(2itheta)" "(3)`

Dividing (3) by (2), we get
`(z_(2)^(2))/(z_(3)) = (cos^(2) thetaz_(1))/(cos 2 theta)`
Hence,`z_(2)^(2) cos 2 theta = z_(1)z_(3) cos^(2) theta`
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