Question

The point $(4,1)$ undergoes the following three transformations successively
I. Reflection about the line $y=x$.
II. Transformation through a distance $2$ units along 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 the point is given by the coordinates

• A. $(\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}})$
• B. $(-\sqrt{2},7\sqrt{2})$
• C. $(-\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}})$
• D. $(\sqrt{2},7\sqrt{2})$

Answer 1

Answer: (C)

Let $B,C,D$ be the position of the point $A(4,1)$ after the three operations I, II and III, respectively. Then, $B$ is $(1,4),C(1+2,4)$ i.e. $(3,4)$. The point $D$ is obtained from $C$ by rotating the coordinate axes through an angle $\pi /4$ in anti-clockwise direction.
Therefore, the coordinates of $D$ are given by
$X=3\cos \frac{\pi }{4}-4\sin \frac{\pi }{4}=-\frac{1}{\sqrt{2}}$
and $Y=3\sin \frac{\pi }{4}+4\cos \frac{\pi }{4}=\frac{7}{\sqrt{2}}$
$\therefore$ Coordinates of $D$ are $\left(-\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}}\right)$.