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The point $P(3,2)$ undergoes the following transformations successively
(i) Reflection about the line $y=x$
(ii) Translation to a distance of 3 units in the positive direction of $X$-axis
(iii) Rotation through an angle $\frac{\pi}{4}$ about the origin in the counter-cloclswise direction
Then, the final position of that point is
Options:
(i) Reflection about the line $y=x$
(ii) Translation to a distance of 3 units in the positive direction of $X$-axis
(iii) Rotation through an angle $\frac{\pi}{4}$ about the origin in the counter-cloclswise direction
Then, the final position of that point is
Solution:
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Verified Answer
The correct answer is:
$(4 \sqrt{2},-\sqrt{2})$
Reflection about the line $y=x$ the coordinates $(3,2$ becomes ( 2,3$)$. On the translation of $(2,3)$ a distance of 3 units with positive direction of $X$-axis the point becomes $(5,3)$. On rotation through on angle $\frac{\pi}{4}$ about origin in the counter-clockwise direction, then coordinate becomes
$$
\begin{aligned}
& \left(5 \cos \frac{\pi}{4}+3 \sin \frac{\pi}{4},-5 \sin \frac{\pi}{4}+3 \cos \frac{\pi}{4}\right) \\
& =\left(\frac{5}{\sqrt{2}}+\frac{3}{\sqrt{2}}, \frac{-5}{\sqrt{2}}+\frac{3}{\sqrt{2}}\right)=(4 \sqrt{2},-\sqrt{2})
\end{aligned}
$$
$$
\begin{aligned}
& \left(5 \cos \frac{\pi}{4}+3 \sin \frac{\pi}{4},-5 \sin \frac{\pi}{4}+3 \cos \frac{\pi}{4}\right) \\
& =\left(\frac{5}{\sqrt{2}}+\frac{3}{\sqrt{2}}, \frac{-5}{\sqrt{2}}+\frac{3}{\sqrt{2}}\right)=(4 \sqrt{2},-\sqrt{2})
\end{aligned}
$$
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