Question

The origin is translated to $(1,2)$. The point $(7,5)$ in the old system undergoes the following transformations successively.
I. Moves to the new point under the given translation of origin.
II. Translated through 2 units along the negative direction of the new $X$ -axis.
III. Rotated through an angle $\frac{\pi }{4}$ about the origin of new system in the clockwise direction. The final position of the point $(7,5)$ is

• A. $\left(\frac{9}{\sqrt{2}},\frac{-1}{\sqrt{2}}\right)$
• B. $\left(\frac{7}{\sqrt{2}},\frac{1}{\sqrt{2}}\right)$
• C. $\left(\frac{7}{\sqrt{2}},\frac{-1}{\sqrt{2}}\right)$
• D. $\left(\frac{5}{\sqrt{2}},\frac{-1}{\sqrt{2}}\right)$

Answer 1

Answer: (C)

Under the translation of origin to $(1,2)$ the point $(7,5)$ undergoes to $(7-1,5-2)\equiv (6,3)$
Under the translation through 2 units along the negative direction of the new $x$ -axis, the point $(6,3)$ undergoes to $(6-2,3)\equiv (4,3)$
Under the rotation throw an angle $\frac{\pi }{4}$ about the origin of new system in the clockwise direction, the final position of point $(7,5)$
$=\left(4\cos \frac{\pi }{4}+3\sin \frac{\pi }{4},-4\sin \frac{\pi }{4}+3\cos \frac{\pi }{4}\right)$
$=\left(\frac{4}{\sqrt{2}}+\frac{3}{\sqrt{2}},-\frac{4}{\sqrt{2}}+\frac{3}{\sqrt{2}}\right)=\left(\frac{7}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right)$