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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
Options:
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
Solution:
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Verified Answer
The correct answer is:
$\left(\frac{7}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$
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
$$
\left(3 \cos \frac{\pi}{4}+4 \sin \frac{\pi}{4}, 4 \cos \frac{\pi}{4}-3 \cos \frac{\pi}{4}\right) \text { i.e. }\left(\frac{7}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)
$$
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
$$
\left(3 \cos \frac{\pi}{4}+4 \sin \frac{\pi}{4}, 4 \cos \frac{\pi}{4}-3 \cos \frac{\pi}{4}\right) \text { i.e. }\left(\frac{7}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)
$$
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