The point P(a, b) undergoes the following three transformations s… - Vidyadip

Question

The point $\mathrm{P}(\mathrm{a}, \mathrm{b})$ 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 $\left(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}\right)$, then the value of $2 a+b$ is equal to:

✓

Answer

a
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

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

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

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

$\text { and } b+2+a=7....(2)$

$\Rightarrow a=4 ; b=1$

$\Rightarrow 2 a+b=9$

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