Question

The point $(4,1)$ undergoes the following transformations successively
I. Reflection about the line $y=x$
II. Translation through a distance $2$ units in the direction of positive $X$ - axis.
III. Rotation through an angle $\frac{\pi}{4}$ about origin in the anticlockwise direction.
Then, the final position of the point is

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

Answer 1

Answer: (A)

We have,
Given point $(4, 1)$.
When $(4,1)$ is reflected about line $y=x$, the new coordinates become $(1,4)$.
Again $(1,4)$ is translated through a distance 2 units in the direction of positive $X$ - axis, the new coordinates become $(1+2,4+0)$ i.e. $(3,4)$.
Lastly $(3,4)$ is rotated through an angle $\frac{\pi}{4}$ about origin in the anticlockwise direction, then new coordinates becomes
$\left(3 \cos \frac{\pi}{4}+4 \sin \frac{\pi}{4}, 4 \cos \frac{\pi}{4}-3 \cos \frac{\pi}{4}\right)$
i.e. $\left(\frac{7}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$