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 given problem, we need to analyze the equation and find the maximum and minimum values of \(\sqrt{\alpha^2 + \beta^2}\) for the ordered pair \((\alpha, \beta)\) that satisfies the equation \(x^2 - 4x + 4y^2 + 3 = 0\).

### Step 1: Rewrite the equation
We start with the equation:
\[ x^2 - 4x + 4y^2 + 3 = 0 \]
We can complete the square for the \(x\) terms:
\[ (x^2 - 4x + 4) + 4y^2 + 3 - 4 = 0 \]
This simplifies to:
\[ (x - 2)^2 + 4y^2 - 1 = 0 \]
Rearranging gives:
\[ (x - 2)^2 + 4y^2 = 1 \]

### Step 2: Identify the shape of the equation
The equation \((x - 2)^2 + 4y^2 = 1\) represents 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 3: Find the coordinates of the ellipse
The points on the ellipse can be parameterized as:
\[ x = 2 + \cos(t), \quad y = \frac{1}{2} \sin(t) \]
for \(t \in [0, 2\pi)\).

### Step 4: Express \(\sqrt{\alpha^2 + \beta^2}\)
We need to find \(\sqrt{\alpha^2 + \beta^2}\):
\[ \sqrt{\alpha^2 + \beta^2} = \sqrt{(2 + \cos(t))^2 + \left(\frac{1}{2} \sin(t)\right)^2} \]
Expanding this gives:
\[ = \sqrt{(2 + \cos(t))^2 + \frac{1}{4} \sin^2(t)} \]
\[ = \sqrt{4 + 4\cos(t) + \cos^2(t) + \frac{1}{4} \sin^2(t)} \]
Using the identity \(\sin^2(t) + \cos^2(t) = 1\), we can simplify:
\[ = \sqrt{4 + 4\cos(t) + \frac{5}{4}} \]
\[ = \sqrt{\frac{16 + 16\cos(t) + 5}{4}} = \frac{1}{2}\sqrt{21 + 16\cos(t)} \]

### Step 5: Find maximum and minimum values
The maximum and minimum values of \(\sqrt{21 + 16\cos(t)}\) occur when \(\cos(t)\) is at its maximum (1) and minimum (-1):
- Maximum: \(\sqrt{21 + 16 \cdot 1} = \sqrt{37}\)
- Minimum: \(\sqrt{21 + 16 \cdot (-1)} = \sqrt{5}\)

Thus, the maximum value \(l\) is:
\[ l = \frac{1}{2}\sqrt{37} \]
And the minimum value \(s\) is:
\[ s = \frac{1}{2}\sqrt{5} \]

### Step 6: Calculate \((l - s)/(l + s)\)
Now we calculate:
\[ l - s = \frac{1}{2}(\sqrt{37} - \sqrt{5}), \quad l + s = \frac{1}{2}(\sqrt{37} + \sqrt{5}) \]
Thus:
\[ \frac{l - s}{l + s} = \frac{\frac{1}{2}(\sqrt{37} - \sqrt{5})}{\frac{1}{2}(\sqrt{37} + \sqrt{5})} = \frac{\sqrt{37} - \sqrt{5}}{\sqrt{37} + \sqrt{5}} \]

### Final Answer
The value of \(\frac{l - s}{l + s}\) is:
\[ \frac{\sqrt{37} - \sqrt{5}}{\sqrt{37} + \sqrt{5}} \]