Question

The point $(4,1)$ undergoes the following three transformations succesively.
$\left(i\right)$Image about the line $y=x$
$\left(ii\right)$Transformation through a distance $2$ unit along the positive direction of $x$ axis
$\left(iii\right)$Rotation through an angle of $\frac{\pi }{4}$ about the origin in the anti-clockwise direction.

The final position of the point is given by the coordinates

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

Answer 1

The correct option is C $(-\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}})$

Image of any point about the line $y=x$ the cooredinates get reversed
Hence, image of point $(4,1)$ in line $x=y$ will be $(1,4)$
Now shifting of the point along $+ivex-$axis direction $(1,4)\to (1+2,4)\equiv (3,4)$
$\left(iii\right)$Now rotation about the $\pi /4$ in anticlockwise
So, $x=X\cos 45{}^{\circ }+Y\sin 45{}^{\circ }\Rightarrow 3\sqrt{2}=X+Y$
$y=-X\sin 45{}^{\circ }+Y\cos 45{}^{\circ }\Rightarrow 4\sqrt{2}=-X+Y$
Solving above equations we get the coordinates of points in new position as $(-\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}})$