Question

Z is rotated through an angle of anticlockwise to get $z_1$ and clockwise to get $z_2$, then

• A. , z, are in GP
• B. , z, are in AP
• C. + = 2 z cos
• D. + = 2 cos(2 )

Answer 1

Answer: (A), (C), (D)

The correct options are
A

, z, are in GP

C

+ = 2 z cos

D

+ = 2 cos(2 )

We will first find the complex number $z_1$ and $z_2$ using the concept of rotation.

$\frac{z_1}{z}$ = $e^{i\alpha }$ and $\frac{z_2}{z}$ = $e^{-i\alpha }$

$\Rightarrow$ $z_1$ = z$e^{i\alpha }$ , $z_2$ = z$e^{-i\alpha }$

$z_1$$z_2$ = z$e^{i\alpha }$ $\times$ z$e^{-i\alpha }$

$\Rightarrow$ $z_1$ ,z ,$z_2$ are in g.p - a

$z_1$ + $z_2$ = z$e^{i\alpha }$ + z$e^{-i\alpha }$

= z ( $e^{i\alpha }$ + $e^{-i\alpha }$ )

= $z(cos\alpha +isin\alpha +cos\alpha -isin\alpha )$

= $2zcos\alpha$ - Option C

$z_1^2$ + $z_2^2$ = $z^2$${\left(e^{-i\alpha }\right)}^2$ + $z^2$${\left(e^{-i\alpha }\right)}^2$

= $z^2$($e^{i2\alpha }$ + $e^{-i2\alpha }$)

= 2 $Z^2$cos 2 $\alpha$ - Option D