Question

The point $P(3,2)$ undergoes the following transformations successively
(i) Reflection about the line $y=x$
(ii) Translation to a distance of $3$ units in the positive direction of $X$ -axis
(iii) Rotation through an angle $\frac{\pi}{4}$ about the origin in the counter-clockwise direction
Then, the final position of that point is

• A. $(2,4)$
• B. $(4 \sqrt{2},-\sqrt{2})$
• C. $\left(\frac{1}{\sqrt{2}}, \sqrt{2}\right)$
• D. $(\sqrt{2}, 2 \sqrt{2})$

Answer 1

Answer: (B)

Reflection about the line $y=x$ the coordinates ($3, 2$ becomes $(2,3)$. On the translation of $(2,3)$ a distance of 3 units with positive direction of $X$ -axis the point becomes $(5,3) .$
On rotation through on angle $\frac{\pi}{4}$ about origin in the counter-clockwise direction, then coordinate becomes
$\left(5 \cos \frac{\pi}{4}+3 \sin \frac{\pi}{4},-5 \sin \frac{\pi}{4}+3 \cos \frac{\pi}{4}\right)$
$=\left(\frac{5}{\sqrt{2}}+\frac{3}{\sqrt{2}}, \frac{-5}{\sqrt{2}}+\frac{3}{\sqrt{2}}\right)$
$=(4 \sqrt{2},-\sqrt{2})$