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. $\Bigg(\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}}\Bigg)$
• B. $(-\sqrt{2}, 7\sqrt{2})$
• C. $\Bigg(-\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}}\Bigg)$
• 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 $\left(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}\right)$.