Question

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

• A. $(-\sqrt{18},\sqrt{18})$
• B. (-2,3)
• C. $(0,\sqrt{18})$
• D. (0,3)

Answer 1

Answer: (C)

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) $X=-x\cos \theta +y\sin \theta$
$=\left(-\frac{3}{\sqrt{2}}+\frac{3}{\sqrt{2}}\right)=0$
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 $(0,\sqrt{18})$.