Question

The point $(4,1)$ undergoes the following three transformations successively:
(I) Reflection about the line $y=x$
(II) Translation through a distance $2$ units along the positive direction of $x$-axis.
(III) Rotation of coordinate axes through an angle $\frac{\pi }{4}$ about the origin in the clockwise direction.
The final position of the point is given by the coordinates

• A. $\left(\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}}\right)$
• B. $(-2,\frac{7}{\sqrt{2}})$
• C. $\left(-\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}}\right)$
• D. $(\sqrt{2},\frac{7}{\sqrt{2}})$

Answer 1

Answer: (C)

$\left(-\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}}\right)$

Let $B,C,D$ be the position of point $A(4,1)$ after the three operations I, II and III respectively.

Then, $B$ is $(1,4)$, $C$ is $(1+2,4)=(3,4)$.

The point $D$ is obtained from V by rotating the coordinate aces through an angle $\frac{\pi }{4}$ in the anticlockwise direction.

Therefore, the coordinate of $D$ is given by

$X=3cos\frac{\pi }{4}-4sin\frac{\pi }{4}=-\frac{1}{\sqrt{2}}Y=3sin\frac{\pi }{4}+4cos\frac{\pi }{4}=\frac{7}{\sqrt{2}}$

Therefore coordinates of $D$ are $(-\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}})$