# Solve this following

Question:

The point $P(a, b)$ undergoes the following three transformations successively :

(a) reflection about the line $y=x$.

(b) translation through 2 units along the positive direction of $\mathrm{x}$-axis.

(c) rotation through angle $\frac{\pi}{4}$ about the origin in the anti-clockwise direction.

If the co-ordinates of the final position of the point $P$ are $\left(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}\right)$, then the value of $2 a+b$ is equal to :

1. 13

2. 9

3. 5

4. 7

Correct Option: , 2

Solution:

Image of $\mathrm{A}(\mathrm{a}, \mathrm{b})$ along $\mathrm{y}=\mathrm{x}$ is $\mathrm{B}(\mathrm{b}, \mathrm{a})$. Translating it 2 units it becomes $C(b+2, a)$.

Now, applying rotation theorem

$-\frac{1}{2}+\frac{7}{\sqrt{2}} \mathrm{i}=((\mathrm{b}+2)+\mathrm{ai})\left(\cos \frac{\pi}{4}+\mathrm{i} \sin \frac{\pi}{4}\right)$

$\frac{-1}{\sqrt{2}}+\frac{7}{\sqrt{2}} \mathrm{i}=\left(\frac{\mathrm{b}+2}{\sqrt{2}}-\frac{\mathrm{a}}{\sqrt{2}}\right)+\mathrm{i}\left(\frac{\mathrm{b}+2}{\sqrt{2}}+\frac{\mathrm{a}}{\sqrt{2}}\right)$

$\Rightarrow b-a+2=-1$    ..............(I)

and $b+2+a=7$                  .............(II)

$\Rightarrow \mathrm{a}=4 ; \mathrm{b}=1$

$\Rightarrow 2 \mathrm{a}+\mathrm{b}=9$