Question

Given a point $A\left(z\right)$ in the Argand plane. $B\left(z^{\prime }\right)$ and $C\left(z^{\prime \prime }\right)$ are points obtained on rotating $A\left(z\right)$ about the Origin through an angle $\alpha$ anticlockwise and clockwise, respectively, without changing modulus of $z$. Identify the correct statement(s).

• A. $z^{\prime },z,z^{\prime \prime }$ form a G.P.
• B. $z^{\prime },z,z^{\prime \prime }$ form an A.P.
• C. $(z^{\prime }{)}^2+(z^{\prime \prime }{)}^2=2z^{2}cos2\alpha$
• D. $z^{\prime }+z^{\prime \prime }=2zcos2\alpha$

Answer 1

Answer: (A), (C)

The correct options are
A $z^{\prime },z,z^{\prime \prime }$ form a G.P.
C $(z^{\prime }{)}^2+(z^{\prime \prime }{)}^2=2z^{2}cos2\alpha$

$z^{\prime }=z{e}^{i\alpha },z^{\prime \prime }=z{e}^{-i\alpha }$

$z^{\prime }{z}^{\prime \prime }=z^2\Rightarrow z^{\prime },z,z^{\prime \prime }$ are in G.P.

${\left(\frac{z^{\prime }}{z}\right)}^2+{\left(\frac{z^{\prime \prime }}{z}\right)}^2=2\cos 2\alpha$
$\Rightarrow (z^{\prime }{)}^2+(z^{\prime \prime }{)}^2=2z^2\cos 2\alpha$

and $z^{\prime }+z^{\prime \prime }=2z\cos \alpha$