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 2a + b is equal to :

Solution

Maths sir · 4 steps

Track the general point (a, b) through each move, then match it with the final point.

1. Reflect, then shift. Reflection in y = x swaps the coordinates, so (a, b) → (b, a). Moving 2 units along the positive x-axis adds 2 to the first coordinate, giving (b + 2, a).
2. Rotate the point. An anticlockwise rotation by θ about the origin sends (x, y) to (x cos θ - y sin θ, x sin θ + y cos θ). With cos π/4 = sin π/4 = 1/√2, this gives the point below.

   ( (b + 2 - a)/√2, (b + 2 + a)/√2 ) = ( -1/√2, 7/√2 )

3. Compare coordinates. So b + 2 - a = -1 and b + 2 + a = 7. Adding the two gives b + 2 = 3, so b = 1, and then a = 4.
4. So. 2a + b = 2(4) + 1 = 9, which is option (b).

The idea to keep. Carry a general point through each move in the given order. Reflection in y = x swaps coordinates; rotation by θ gives (x cos θ - y sin θ, x sin θ + y cos θ).

Answer from the NTA final answer key. Solution by PrepWiser.