The point P(a,b) undergoes reflection about y=x, then translation 2 units along positive X…

Question

The point $P(a,b)$ undergoes reflection about $y=x$, then translation 2 units along positive $X$-axis, then rotation through angle $\dfrac\pi4$ about the origin anticlockwise. If the final position is $\left(-\dfrac1{\sqrt2},\dfrac7{\sqrt2}\right)$, then the value of $2a+b$ is equal to

Accepted Answer

Reflecting $(a,b)\to(b,a)$, translating $\to(b+2,a)$, then rotating by $e^{i\pi/4}$:

$$-\frac1{\sqrt2}+\frac7{\sqrt2}i=[(b+2)+ai]\left(\cos\frac\pi4+i\sin\frac\pi4\right)=\frac{(b-a+2)}{\sqrt2}+i\,\frac{(a+b+2)}{\sqrt2}.$$

So $b-a+2=-1 \implies b-a=-3$, and $a+b+2=7 \implies a+b=5$. Solving: $b=1,\ a=4$.

$2a+b=8+1=9.$

Solutions · 1