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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 $\mathrm{x}-axis.$
(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
Option: 1 $13$
Option: 2 $9$
Option: 3 $5$
Option: 4 $7$

Answers (1)

best_answer

$1) Reflection \: \: along \: \: y=x \: \: will \: \: make\: \: the \: \: point\, (b, a)$

$2) 2 \: \: units\: \: in +x$ direction: $(b+2, a)$

$3)$

$Using \: \: Rotation \: \: Theorem$

$\begin{aligned} & \frac{-\frac{1}{\sqrt{2}}+\frac{7}{\sqrt{2}} i}{b+2+a i}=e^{i \pi / 4} \\ \Rightarrow &-\frac{1}{\sqrt{2}}+\frac{7}{\sqrt{2}} i=((b+2)+a i)\left(\frac{1}{\sqrt{2}}+i \cdot \frac{1}{\sqrt{2}}\right) \\ \Rightarrow &-\frac{1}{\sqrt{2}}+\frac{7}{\sqrt{2}} i=\frac{b+2-a}{\sqrt{2}}+i\left(\frac{b+2+a}{\sqrt{2}}\right) \end{aligned}$

$Comparing$

$b+2-a=-1 \: and \: b+2+a=7$

$\Rightarrow a=4, b=1$

$\Rightarrow 2 a+b=9$

The correct option is (2)

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Deependra Verma

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