Question

If a point in argand plane $A(2,3)$ rotated through origin by $\frac{\pi }{4}$ in anticlockwise direction, then the new coordinates of the point will be

• A. $(\frac{1}{\sqrt{2}},-\frac{5}{\sqrt{2}})$
• B. $(-\frac{1}{\sqrt{2}},-\frac{5}{\sqrt{2}})$
• C. $(\frac{1}{\sqrt{2}},\frac{5}{\sqrt{2}})$
• D. $(-\frac{1}{\sqrt{2}},\frac{5}{\sqrt{2}})$

Answer 1

The correct option is D $(-\frac{1}{\sqrt{2}},\frac{5}{\sqrt{2}})$

Let $z_A=2+3i$ and after rotation the complex number be $z_B$ then
$|z_A|=|z_B|$
Using rotation property we have
$\frac{z_B}{z_A}=\frac{|z_B|}{|z_A|}{e}^{i\pi /4}\Rightarrow z_B=z_A{e}^{i\pi /4}\Rightarrow z_B=(2+3i)(\frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}})\therefore z_B=-\frac{1}{\sqrt{2}}+\frac{5}{\sqrt{2}}i$
Hence, the required point will be
$(-\frac{1}{\sqrt{2}},\frac{5}{\sqrt{2}})$