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 directions of x-axis.
(iii) Rotation through an angle $\frac{\pi }{4}$ about the origin.
The final position of the point is given by the coordinates:

• A. $\left(\frac{7}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right)$
• B. $\left(\frac{7}{\sqrt{2}},\frac{1}{\sqrt{2}}\right)$
• C. $\left(-\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}}\right)$
• D. None

Answer 1

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

New position of point $(4,1)$ after reflection in $y=x$ is $(1,4)$

After translation it will be $(1+2,4)=(3,4)$

After rotation at ${45}^o$ anticlockwise it will be

$(3\cos {45}^o-4\sin {45}^o,3\sin {45}^o+4\cos {45}^o)$

$=(3\times \frac{1}{\sqrt{2}}-4\times \frac{1}{\sqrt{2}},\frac{3}{\sqrt{2}}+\frac{4}{\sqrt{2}})$

$=(\frac{-1}{\sqrt{2}},\frac{7}{\sqrt{2}})$.