Let `(alpha, beta)` be an ordered pair of real numbers satisfying the equation `x^(2)-4x+4y^(2)+3=0`. If the maximum and minimum value of `sqrt(alpha^(2)+beta^(2))` are l and s respectively, then the value of `(l-s)/(l+s)` is

Answer

To solve the problem, we need to analyze the given equation and find the maximum and minimum distances from the origin to the points \((\alpha, \beta)\) that satisfy the equation.

### Step 1: Rewrite the given equation
The equation given is:
\[ x^2 - 4x + 4y^2 + 3 = 0 \]
We can rearrange it to isolate the terms involving \(x\) and \(y\):
\[ x^2 - 4x + 4y^2 = -3 \]

### Step 2: Complete the square for the \(x\) terms
To complete the square for the \(x\) terms:
\[ x^2 - 4x = (x - 2)^2 - 4 \]
Substituting this back into the equation gives:
\[ (x - 2)^2 - 4 + 4y^2 = -3 \]
This simplifies to:
\[ (x - 2)^2 + 4y^2 = 1 \]

### Step 3: Identify the standard form of the ellipse
The equation \((x - 2)^2 + 4y^2 = 1\) can be rewritten as:
\[ \frac{(x - 2)^2}{1} + \frac{y^2}{\frac{1}{4}} = 1 \]
This is the standard form of an ellipse centered at \((2, 0)\) with semi-major axis \(1\) (along the \(x\)-axis) and semi-minor axis \(\frac{1}{2}\) (along the \(y\)-axis).

### Step 4: Find the distances from the origin to the ellipse
The distance from the origin \((0, 0)\) to a point \((\alpha, \beta)\) on the ellipse can be expressed as:
\[ d = \sqrt{\alpha^2 + \beta^2} \]
To find the maximum and minimum distances, we need to evaluate the distance from the origin to the ellipse.

### Step 5: Determine the minimum distance
The closest point on the ellipse to the origin occurs when the line connecting the origin to the center of the ellipse \((2, 0)\) is extended to the edge of the ellipse. The minimum distance occurs at the point \((1, 0)\):
\[ \text{Minimum distance} = \sqrt{1^2 + 0^2} = 1 \]

### Step 6: Determine the maximum distance
The farthest point on the ellipse from the origin occurs at the point \((3, 0)\):
\[ \text{Maximum distance} = \sqrt{3^2 + 0^2} = 3 \]

### Step 7: Assign values to \(l\) and \(s\)
Let:
- \(l = 3\) (maximum distance)
- \(s = 1\) (minimum distance)

### Step 8: Calculate \((l - s)/(l + s)\)
Now we can calculate:
\[ \frac{l - s}{l + s} = \frac{3 - 1}{3 + 1} = \frac{2}{4} = \frac{1}{2} \]

### Final Answer
The value of \(\frac{l - s}{l + s}\) is:
\[ \frac{1}{2} \]