If the area (in sq. units) of the region `{(x,y):y^(2)le 4x, x+y le 1, x ge 0, y ge 0}` is `a sqrt(2) +b`, then `a-b` is equal to :

Options

A. `(8)/(3)`
B. `(10)/(3)`
C. `-(2)/(3)`
D. 6

Answer

To solve the problem of finding the area of the region defined by the inequalities \(y^2 \leq 4x\), \(x + y \leq 1\), \(x \geq 0\), and \(y \geq 0\), we will follow these steps:

### Step 1: Identify the curves and intersection points
The equation \(y^2 = 4x\) represents a parabola that opens to the right. The line \(x + y = 1\) can be rewritten as \(y = 1 - x\). We need to find the points where these two curves intersect.

1. Set \(y^2 = 4x\) and substitute \(y = 1 - x\):
\[ (1 - x)^2 = 4x \]
Expanding this gives:
\[ 1 - 2x + x^2 = 4x \]
Rearranging leads to:
\[ x^2 - 6x + 1 = 0 \]

### Step 2: Solve for \(x\)
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[ x = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{6 \pm \sqrt{36 - 4}}{2} = \frac{6 \pm \sqrt{32}}{2} = \frac{6 \pm 4\sqrt{2}}{2} = 3 \pm 2\sqrt{2} \]
Thus, the \(x\)-coordinates of the intersection points are \(3 + 2\sqrt{2}\) and \(3 - 2\sqrt{2}\).

### Step 3: Find the corresponding \(y\)-coordinates
Substituting \(x = 3 - 2\sqrt{2}\) back into \(y = 1 - x\):
\[ y = 1 - (3 - 2\sqrt{2}) = 2\sqrt{2} - 2 \]
So the intersection point is \((3 - 2\sqrt{2}, 2\sqrt{2} - 2)\).

### Step 4: Determine the area
The area we need to find is bounded by the parabola and the line. We can find the area by integrating the difference between the line and the parabola from \(x = 0\) to \(x = 1\).

1. The area under the line from \(x = 0\) to \(x = 1\):
\[ A_{\text{line}} = \int_0^1 (1 - x) \, dx = \left[x - \frac{x^2}{2}\right]_0^1 = 1 - \frac{1}{2} = \frac{1}{2} \]

2. The area under the parabola from \(x = 0\) to \(x = 1\):
\[ A_{\text{parabola}} = \int_0^{1} \frac{y^2}{4} \, dy = \int_0^{1} \frac{y^2}{4} \, dy = \frac{1}{4} \cdot \left[\frac{y^3}{3}\right]_0^{2\sqrt{2} - 2} = \frac{(2\sqrt{2} - 2)^3}{12} \]

### Step 5: Calculate the total area
The total area is given by:
\[ \text{Area} = A_{\text{line}} - A_{\text{parabola}} \]
We will need to evaluate the integrals and simplify to find the final area in the form \(a\sqrt{2} + b\).

### Step 6: Find \(a - b\)
After calculating the area, we will express it in the form \(a\sqrt{2} + b\) and find \(a - b\).