Q. The point (4, 1) undergoes the following three transformations successively (i) Refletion about the line $y = x$ (ii) Translation through a distance of 2 unit along the positive direction of $x$-axis (iii) Rotation through an angle of $\frac{\pi}{4}$ about the origin in the anti-clockwise direction The final position of the point is
A) $\left( \frac{1}{\sqrt{2}},\ \frac{7}{\sqrt{2}} \right)$
B) $\left( - \frac{1}{\sqrt{2}},\ \frac{7}{\sqrt{2}} \right)$
C) $( - \sqrt{2},\ 7\sqrt{2})$
D) $(\sqrt{2},\ 7\sqrt{2})$
Explanation
(b)
Let $z = 4 + i$ when reflected along $y = x$ will become $z = 1 + 4i$
When translated by 2 unit $z = 3 + 4i$
When rotated by angle $\pi/4$ in anti-clockwise direction will give
$z = (3 + 4i)\left( \cos{\frac{\pi}{4} + i\sin\frac{\pi}{4}} \right)$
$z = \frac{1}{\sqrt{2\ }}\left\lbrack 3 - 4 + i(3 + 4) \right\rbrack = - \frac{1}{\sqrt{2}} + i\frac{7}{\sqrt{2}}\$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.