JEE Main 2021 (Online) 27th July Evening ShiftQuestion 1
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 $\frac\pi4$ about the origin in the anti-clockwise direction. If the co-ordinates of the final position of the point $P$ are $\left(-\frac1{\sqrt2},\frac7{\sqrt2}\right)$, then the value of $2a+b$ is equal to :
Points and Straight Lines
Solution
Rotation by $\frac\pi4$: $(x,y)\to\left(\frac{x-y}{\sqrt2},\frac{x+y}{\sqrt2}\right)=\left(-\frac1{\sqrt2},\frac7{\sqrt2}\right)$, so $x-y=-1,\ x+y=7\Rightarrow(x,y)=(3,4)$. Undo the translation: $(1,4)$. Undo the reflection about $y=x$: $P=(4,1)$. $2a+b=9$. Answer: (C).