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The point $(3,2)$ undergoes the following three transformations in the order given
(i) Reflection about the line $y=x$.
(ii) Translation by the distance 1 unit in the positive direction of $x$-axis.
(iii) Rotation by an angle $\frac{\pi}{4}$ about the origin in the anti-clockwise direction.
Then, the final position of the point is
Options:
(i) Reflection about the line $y=x$.
(ii) Translation by the distance 1 unit in the positive direction of $x$-axis.
(iii) Rotation by an angle $\frac{\pi}{4}$ about the origin in the anti-clockwise direction.
Then, the final position of the point is
Solution:
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Verified Answer
The correct answer is:
$(0, \sqrt{18})$
Given point is $(3,2)$.
(i) Reflection of point $(3,2)$ about the line $y=x$ is $(2,3)$.
(ii) Translation of a point through 1 unit distance in the positive direction of $x$-axis is $(3,3)$.
(iii)
$\begin{aligned} X & =-x \cos \theta+y \sin \theta \\ & =\left(-\frac{3}{\sqrt{2}}+\frac{3}{\sqrt{2}}\right)=0\end{aligned}$
and $Y=x \sin \theta+y \cos \theta$
$=\left(\frac{3}{\sqrt{2}}+\frac{3}{\sqrt{2}}\right)=3 \sqrt{2}=\sqrt{18}$
Hence, final position is $(0, \sqrt{18})$.
(i) Reflection of point $(3,2)$ about the line $y=x$ is $(2,3)$.
(ii) Translation of a point through 1 unit distance in the positive direction of $x$-axis is $(3,3)$.
(iii)
$\begin{aligned} X & =-x \cos \theta+y \sin \theta \\ & =\left(-\frac{3}{\sqrt{2}}+\frac{3}{\sqrt{2}}\right)=0\end{aligned}$
and $Y=x \sin \theta+y \cos \theta$
$=\left(\frac{3}{\sqrt{2}}+\frac{3}{\sqrt{2}}\right)=3 \sqrt{2}=\sqrt{18}$
Hence, final position is $(0, \sqrt{18})$.
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