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 π 4 about the origin in the anti-clockwise direction. If the co-ordinates of the final position of the point P are - 1 2 , 7 2 , then the value of 2 a + b is equal to:

Step-by-step solution

Image of A a , b along y = x is P ' b , a . Translating it 2 units along positive direction of x - axis it becomes P " b + 2 , a . Now, applying Rotation theorem - 1 2 + 7 2 i = b + 2 + a i cos π 4 + i sin π 4 Multiplying the complex numbers we get, - 1 2 + 7 2 i = b + 2 2 - a 2 + i b + 2 2 + a 2 ⇒ b - a + 2 = - 1           . . . 1 and b + 2 + a = 7             . . . 2 ⇒ a = 4 ; b = 1 ⇒ 2 a + b = 9