The minimum value of `[x_1-x_2)^2 + ( 12 - sqrt(1 - (x_1)^2)- sqrt(4 x_2)]^(1/2)` for all permissible values of `x_1` and `x_2` is equal to `asqrtb -c` where `a,b,c in N`, the find the value of a+b-c

Answer

To find the minimum value of the expression \((x_1 - x_2)^2 + \sqrt{12 - \sqrt{1 - (x_1)^2} - \sqrt{4x_2}}\), we will follow these steps:

### Step 1: Identify the components of the expression
The expression consists of two parts:
1. The squared difference \((x_1 - x_2)^2\)
2. The square root term \(\sqrt{12 - \sqrt{1 - (x_1)^2} - \sqrt{4x_2}}\)

### Step 2: Analyze the square root term
The term inside the square root must be non-negative for the expression to be defined. Therefore, we need:
\[ 12 - \sqrt{1 - (x_1)^2} - \sqrt{4x_2} \geq 0 \]
This implies:
\[ \sqrt{4x_2} \leq 12 - \sqrt{1 - (x_1)^2} \]

### Step 3: Simplify the square root condition
Squaring both sides gives:
\[ 4x_2 \leq (12 - \sqrt{1 - (x_1)^2})^2 \]
Expanding the right-hand side:
\[ 4x_2 \leq 144 - 24\sqrt{1 - (x_1)^2} + (1 - (x_1)^2) \]
This simplifies to:
\[ 4x_2 \leq 145 - (x_1)^2 - 24\sqrt{1 - (x_1)^2} \]

### Step 4: Set up the distance formula
We can interpret the expression as a distance from a point to a curve. The first term \((x_1 - x_2)^2\) represents the horizontal distance, while the square root term represents a vertical distance.

### Step 5: Find the minimum distance
To minimize the entire expression, we can consider the geometry of the problem. The point \((x_1, y_1)\) lies on a parabola and the point \((x_2, y_2)\) lies on a circle. We need to find the shortest distance between these two curves.

### Step 6: Determine the coordinates of the curves
The parabola can be represented as \(y = 12 - \sqrt{1 - (x_1)^2}\) and the circle as \(x^2 + (y - 12)^2 = 1\).

### Step 7: Find intersection points
To find the intersection points of the parabola and the circle, we can set the equations equal to each other and solve for \(x\) and \(y\).

### Step 8: Calculate the distance
After finding the intersection points, we can calculate the distance from the center of the circle to the point on the parabola. The center of the circle is \((0, 12)\) and the radius is \(1\).

### Step 9: Substitute values to find the minimum distance
Using the coordinates of the intersection points, we can substitute back into the distance formula and simplify to find the minimum value.

### Step 10: Final expression
After simplification, we find that the minimum value can be expressed in the form \(a\sqrt{b} - c\).

### Conclusion
From the final expression, we can identify the values of \(a\), \(b\), and \(c\) and compute \(a + b - c\).