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 $x-$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 $(-\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}})$, then the value of $2a+b$ is equal to

• A. $9$
• B. $5$
• C. $13$
• D. $7$

Answer 1

The correct option is A $9$

Reflection of $P(a,b)$ about line $y=x$ is $P^{\prime }=(b,a)$. After translation of $2$ units the new coordinates are $P^{\prime \prime }=(b+2,a)$
After rotating $\frac{\pi }{4}$let the new coordinates be $(x_1,y_1).$
$\therefore \frac{(x_1+i{y}_1)-0}{(b+2+ai)-0}=e^{i\frac{\pi }{4}}$
$x_1+i{y}_1=\left(\right(b+2)+ai)(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}i)$
$=\frac{1}{\sqrt{2}}(b+2+(b+2)i+ai-a)$
$=\frac{1}{\sqrt{2}}\left(\right(a+b+2)i+(b-a+2\left)\right)$
$\therefore b-a+2=-1,a+b+2=7$
$\therefore a=4,b=1$