The point $(3,2)$ undergoes the following three transformations in the order given (i) Reflection about the…
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
$(-\sqrt{18}, \sqrt{18})$
$(-2,3)$
$(0, \sqrt{18})$
$(0,3)$
Solution
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 $(0, \sqrt{18})$.