Key Concepts and Formulas:

• Reflection about the line $y=x$: The reflection of a point $(x, y)$ about the line $y=x$ is $(y, x)$.
• Translation: A translation by $h$ units along the x-axis and $k$ units along the y-axis maps $(x, y)$ to $(x+h, y+k)$.
• Rotation about the origin: Rotation by an angle $\theta$ counter-clockwise about the origin is efficiently handled using complex numbers. If $z = x + iy$ represents a point, the rotated point $z'$ is given by $z' = z e^{i\theta}$, where $e^{i\theta} = \cos \theta + i \sin \theta$.

Step-by-Step Solution:

1. Reflection about the line y = x

• Concept Applied: Reflection about $y=x$
• Explanation: Reflecting the point $(a, b)$ about the line $y=x$ simply swaps the x and y coordinates.
• Working: The point P(a, b) becomes P₁(b, a). $P(a, b) \xrightarrow{\text{Reflection about } y=x} P_1(b, a)$

2. Translation by 2 units along the positive x-axis

• Concept Applied: Translation.
• Explanation: We add 2 to the x-coordinate of the point.
• Working: The point P₁(b, a) becomes P₂(b+2, a). $P_1(b, a) \xrightarrow{\text{Translation: } x \to x+2} P_2(b+2, a)$

3. Rotation by π/4 about the origin in the counter-clockwise direction

• Concept Applied: Rotation using complex numbers.
• Explanation: Represent the point P₂(b+2, a) as a complex number $z = (b+2) + ai$. Rotating this point by $\theta = \frac{\pi}{4}$ counter-clockwise gives a new complex number $z' = z e^{i\theta}$, where $e^{i\theta} = \cos \theta + i \sin \theta$.
• Working:
  • $z = (b+2) + ai$
  • $\theta = \frac{\pi}{4}$
  • $e^{i{\pi \over 4}} = \cos \left({\pi \over 4}\right) + i \sin \left({\pi \over 4}\right) = \frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}}$
  • $z' = z \cdot e^{i{\pi \over 4}} = \left( (b+2) + ai \right) \left( \frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}} \right)$
  • $z' = \frac{1}{\sqrt{2}} \left( (b+2) + ai \right) (1 + i)$
  • $z' = \frac{1}{\sqrt{2}} \left( (b+2) + (b+2)i + ai - a \right)$
  • $z' = \frac{1}{\sqrt{2}} \left( (b+2-a) + (b+2+a)i \right)$ Therefore, the final coordinates are $\left( \frac{b+2-a}{\sqrt{2}}, \frac{b+2+a}{\sqrt{2}} \right)$.

4. Equating the final coordinates with the given coordinates

• Explanation: We are given that the final coordinates are $\left( { - {1 \over {\sqrt 2 }},{7 \over {\sqrt 2 }}} \right)$. We equate these with the coordinates we derived in the previous step.
• Working: $\frac{b+2-a}{\sqrt{2}} = - \frac{1}{\sqrt{2}}$ $\frac{b+2+a}{\sqrt{2}} = \frac{7}{\sqrt{2}}$ This gives us the following system of equations: $b+2-a = -1 \implies b - a = -3 \quad \text{(Equation 1)}$ $b+2+a = 7 \implies b + a = 5 \quad \text{(Equation 2)}$

5. Solving the system of equations

• Explanation: Solve the two equations for $a$ and $b$.
• Working: Adding Equation (1) and Equation (2): $(b-a) + (b+a) = -3 + 5$ $2b = 2$ $b = 1$ Substituting $b=1$ into Equation (2): $1 + a = 5$ $a = 4$ Therefore, $a = 4$ and $b = 1$.

6. Calculating the value of 2a + b

• Explanation: Substitute the values of $a$ and $b$ into the expression $2a + b$.
• Working: $2a + b = 2(4) + 1 = 8 + 1 = 9$

Tips and Common Mistakes to Avoid:

• Order matters: Always perform transformations in the specified order.
• Complex number rotation: Using complex numbers for rotation simplifies the calculations and reduces the chance of errors.
• Sign errors: Be careful with signs when expanding and simplifying expressions, especially with complex numbers.

Summary: We applied a series of geometric transformations to the point P(a, b). First, we reflected the point about the line y = x, then translated it 2 units along the positive x-axis, and finally rotated it by π/4 about the origin. By equating the final coordinates with the given coordinates, we obtained a system of equations that allowed us to solve for a and b. Finally, we calculated the value of 2a + b, which is 9.

The final answer is \boxed{9}, which corresponds to option (B).