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 {\pi \over 4} about the origin in the anti-clockwise direction. If the co-ordinates

Question

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 {\pi \over 4} about the origin in the anti-clockwise direction. If the co-ordinates of the final position of the point P are \left( { - {1 \over {\sqrt 2 }},{7 \over {\sqrt 2 }}} \right), then the value of 2a + b is equal to :

Accepted Answer

Answer: B. 9 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 - {1 \over {\sqrt 2 }} + {7 \over {\sqrt 2 }}i = \left( {(b + 2) + ai} \right)\left( {\cos {\pi \over 4} + i\sin {\pi \over 4}} \right) - {1 \over {\sqrt 2 }} + {7 \over {\sqrt 2 }}i = \left( {{{b + 2} \over {\sqrt 2 }} - {a \over {\sqrt 2 }}} \right) + i\left( {{{b + 2} \over {\sqrt 2 }} + {a \over {\sqrt 2 }}} \right) \Rightarrow b - a + 2 = -1 ......(i) and b + 2 + a = 7 ...... (ii) \Rightarrow a = 4; b = 1 \Rightarrow 2a + b = 9

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