Question

A complex number z is rotated in anticlockwise direction by an angle $\alpha$ and we get z' and if the same complex number z is rotated by an angle $\alpha$ in clockwise direction and we get z", then

• A. $z^{\prime },z,z^{\prime \prime }$ are in G.P.
• B. z', z, z" are in H.P.
• C. $z^{\prime }+z^{\prime \prime }=2zcos\alpha$
• D. $z^{\prime 2}+z^{\prime \prime 2}=2z^{2}cos2\alpha$

Answer 1

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

The correct options are
A $z^{\prime },z,z^{\prime \prime }$ are in G.P.
C $z^{\prime }+z^{\prime \prime }=2zcos\alpha$
D $z^{\prime 2}+z^{\prime \prime 2}=2z^{2}cos2\alpha$

$z^{\prime }=z{e}^{i\alpha }$ (1)
$z^{\prime \prime }=z{e}^{-i\alpha }$ (2)
$\therefore z^{\prime }{z}^{\prime \prime }=z^2$
$\Rightarrow z^{\prime },z,z^{\prime \prime }$ are in G.P.
Also, ${\left(\frac{z^{\prime }}{z}\right)}^2+{\left(\frac{z^{\prime \prime }}{z}\right)}^2=2cos2\alpha$
$\Rightarrow z^{\prime 2}+z^{\prime \prime 2}=2z^{2}cos2\alpha$
$=2z^2(2co{s}^2\alpha -1)$
or $z^{\prime 2}+z^{\prime \prime 2}+2z^2=4z^{2}co{s}^2\alpha$
or $z^{\prime 2}+z^{\prime \prime 2}+2z^{\prime }{z}^{\prime \prime }=4z^{2}co{s}^2\alpha$
or $(z^{\prime }+z^{\prime \prime }{)}^2=4z^{2}co{s}^2\alpha$
or $z^{\prime }+z^{\prime \prime }=2zcos\alpha$