© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Reflection about the line y = x
A reflection of the point P with coordinates (a, b) about the line
y = x swaps its coordinates.
Thus, after reflection, the new coordinates become
(b, a).
Step 2: Translation along the positive x-axis
The reflected point (b, a) is translated 2 units along the positive x-axis.
This means we add 2 to the x-coordinate.
Hence, the coordinates of the translated point are
(b + 2, a).
Step 3: Rotation by π/4 about the origin
Next, the point (b + 2, a) is rotated about the origin through an angle
$ \frac{\pi}{4} $ in the anti-clockwise direction. One way to handle this rotation
is to interpret the coordinates as a complex number
$ (b + 2) + a\,i $ and then multiply by
$ \cos\left(\frac{\pi}{4}\right) + i \,\sin\left(\frac{\pi}{4}\right)$.
Recall that
$ \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} $
and
$ \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$.
Hence, the product in complex form is:
$ (b + 2 + a\,i)\times \left(\frac{1}{\sqrt{2}} + i\,\frac{1}{\sqrt{2}}\right)\,. $
Multiplying out:
$ (b + 2)\cdot\frac{1}{\sqrt{2}} - a\cdot\frac{1}{\sqrt{2}} + i\left[(b + 2)\cdot\frac{1}{\sqrt{2}} + a\cdot\frac{1}{\sqrt{2}}\right]\,. $
This simplifies to
$ \left(\frac{b + 2 - a}{\sqrt{2}}\right) + i \left(\frac{b + 2 + a}{\sqrt{2}}\right). $
Step 4: Equate to the given final coordinates
The final coordinates of the point are given as
$ \left(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}\right), $
which corresponds to the complex number:
$ -\frac{1}{\sqrt{2}} + i\,\frac{7}{\sqrt{2}}. $
Equating real and imaginary parts, we get the system:
Real part:
$ \frac{b + 2 - a}{\sqrt{2}} = -\frac{1}{\sqrt{2}} \quad\Longrightarrow\quad b + 2 - a = -1. $
Imaginary part:
$ \frac{b + 2 + a}{\sqrt{2}} = \frac{7}{\sqrt{2}} \quad\Longrightarrow\quad b + 2 + a = 7. $
Step 5: Solve for a and b
From the real part equation:
$ b - a + 2 = -1 \quad\Longrightarrow\quad b - a = -3. \quad (1)
$
From the imaginary part equation:
$ b + a + 2 = 7 \quad\Longrightarrow\quad b + a = 5. \quad (2)
$
Adding equations (1) and (2):
$ (b - a) + (b + a) = -3 + 5 \quad\Longrightarrow\quad 2b = 2 \quad\Longrightarrow\quad b = 1.
$
Substituting $ b = 1 $ into equation (2) gives:
$ 1 + a = 5 \quad\Longrightarrow\quad a = 4.
$
Step 6: Find the required expression 2a + b
Now that $ a = 4 $ and $ b = 1 $,
$ 2a + b = 2(4) + 1 = 8 + 1 = 9.
Final Answer:
9
