Question

The point $A(4,1)$ undergoes following transformations successively:(i) reflection about the line $y=x$(ii) translation through a distance of $3units$ in the positive direction of $x-axis$(iii) rotation through an angle ${105}^o$ in anti-clockwise direction about origin $O$. Then the final position of point $A$ is

• A. $(\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}})$
• B. $(-2,7\sqrt{2})$
• C. $(-\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}})$
• D. $(-2\sqrt{6},2\sqrt{2})$

Answer 1

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

Step 1 Reflection in line $y=x$

Point becomes $(1,4)$

Step 2. $3$ units translation in positive $x$ direction

Point becomes $(4,4)$

Step 3. Rotation ${105}^{\circ }$ in anticlockwise direction

$X=x\cos \alpha -y\sin \alpha X=4\cos {105}^{\circ }-4\sin {105}^{\circ }X=4(\cos {105}^{\circ }-\sin {105}^{\circ })=-\frac{1}{\sqrt{2}}Y=x\cos \alpha -y\sin \alpha Y=4\cos {105}^{\circ }+4\sin {105}^{\circ }X=4(\cos {105}^{\circ }+\sin {105}^{\circ })=\frac{7}{\sqrt{2}}(-\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}})$