The point P(a, b) undergoes the following three transformation successively :(a) reflection about the line y = x(b) translation through 2 units along the positive direction of x -axis(c) rotation through angle about the origin in the anti-clockwise directionIf the co-ordinates of the final position of the point P are , then the value of 2a + b is equal to :

Engineering

Mathematics

Translation of Axes

Question

The point P(a, b) undergoes the following three transformation successively :

(a) reflection about the line y = x

(b) translation through 2 units along the positive direction of x -axis

If the co-ordinates of the final position of the point P are $(- \frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}})$ , then the value of 2a + b is equal to :

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Using rotation theorem

$\frac{- 1}{\sqrt{2}} + \frac{7}{\sqrt{2}} i = {(b + 2) + ai} {\cos \frac{π}{4} + isin \frac{π}{4}}$

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

b – a + 2 = – 1 ……. (i)

b + 2 + a = 7 …… (ii)

Using equation (i) & (ii)

a = 4, b = 1

⇒ 2a + b = 9