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 anticlockwise 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 $2a+b$ is equal to :

• A. 13
• B. 9
• C. 5
• D. 7

Answer 1

Answer: (B)

Image of $A(a,b)$ along $y=x$ is $B(b,a)$.
Translating it $2$ units it becomes $C(b+2,a)$
Now, applying rotation theorem
$-\frac{1}{2}+\frac{7}{\sqrt{2}}i=((b+2)+ai)\left(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4}\right)$
$\frac{-1}{\sqrt{2}}+\frac{7}{\sqrt{2}}i=\left(\frac{b+2}{\sqrt{2}}-\frac{a}{\sqrt{2}}\right)+i\left(\frac{b+2}{\sqrt{2}}+\frac{a}{\sqrt{2}}\right)$
$\Rightarrow b-a+2=-1\dots$ (i)
and $b+2+a=7\dots$ (ii)
$\Rightarrow a=4;b=1$
$\Rightarrow 2a+b=9$