Given ` x, y ``in R , x^(2) + y^(2) gt 0 ` . Then the range of `(x^(2) + y^(2))/(x^(2) + xy + 4y^(2))` is (a) `((10 - 4 sqrt(5))/(3),(10 + 4 sqrt(5))/(3))` (b) `((10 - 4 sqrt(5))/(15),(10 + 4 sqrt(5))/(15))` (c) `((5- 4 sqrt(5))/(15),(5 + 4 sqrt(5))/(15))` (d) `((20- 4 sqrt(5))/(15),(20 + 4 sqrt(5))/(15))`

Options

A. `((10 - 4 sqrt(5))/(3).(10 + 4 sqrt(5))/(3))`
B. `((10 - 4 sqrt(5))/(15).(10 + 4 sqrt(5))/(15))`
C. `((5- 4 sqrt(5))/(15).(5 + 4 sqrt(5))/(15))`
D. `((20- 4 sqrt(5))/(15).(20 + 4 sqrt(5))/(15))`

Explanation

To find the range of the expression \(\frac{x^2 + y^2}{x^2 + xy + 4y^2}\), we will follow these steps:

### Step 1: Define the expression
Let \(\lambda = \frac{x^2 + y^2}{x^2 + xy + 4y^2}\).

### Step 2: Introduce a substitution
We can simplify our calculations by introducing a substitution. Let \(z = \frac{y}{x}\). Then we can express \(y\) in terms of \(z\) and \(x\):
- \(y = zx\)

### Step 3: Rewrite the expression in terms of \(z\)
Substituting \(y = zx\) into \(\lambda\):
\[ \lambda = \frac{x^2 + (zx)^2}{x^2 + x(zx) + 4(zx)^2} \]
This simplifies to:
\[ \lambda = \frac{x^2 + z^2x^2}{x^2 + zx^2 + 4z^2x^2} \]
Factoring out \(x^2\):
\[ \lambda = \frac{1 + z^2}{1 + z + 4z^2} \]

### Step 4: Analyze the expression
Now we have:
\[ \lambda = \frac{1 + z^2}{1 + z + 4z^2} \]
To find the range of \(\lambda\), we need to ensure that the denominator is not zero and that \(z\) can take any real value.

### Step 5: Set up the equation for real \(z\)
For \(\lambda\) to be real, we will set up the equation:
\[ \lambda(1 + z + 4z^2) = 1 + z^2 \]
Rearranging gives:
\[ 4\lambda z^2 + \lambda z + (\lambda - 1) = 0 \]
This is a quadratic equation in \(z\).

### Step 6: Apply the discriminant condition
For \(z\) to be real, the discriminant of this quadratic must be non-negative:
\[ D = b^2 - 4ac = (\lambda)^2 - 4(4\lambda)(\lambda - 1) \geq 0 \]
Calculating the discriminant:
\[ D = \lambda^2 - 16\lambda^2 + 16\lambda = -15\lambda^2 + 16\lambda \]
Setting the discriminant \(D \geq 0\):
\[ -15\lambda^2 + 16\lambda \geq 0 \]
Factoring out \(-\lambda\):
\[ \lambda(16 - 15\lambda) \geq 0 \]
This gives us two critical points: \(\lambda = 0\) and \(\lambda = \frac{16}{15}\).

### Step 7: Determine the intervals
The inequality \(\lambda(16 - 15\lambda) \geq 0\) holds when:
1. \(\lambda \leq 0\) or
2. \(\lambda \geq \frac{16}{15}\)

However, since \(\lambda\) is positive (as \(x^2 + y^2 > 0\)), we focus on the second interval.

### Step 8: Find the maximum and minimum values
To find the maximum and minimum values, we need to evaluate the critical points. Solving the quadratic equation gives us:
\[ \lambda = \frac{10 - 4\sqrt{5}}{15} \quad \text{and} \quad \lambda = \frac{10 + 4\sqrt{5}}{15} \]

### Conclusion
Thus, the range of \(\lambda\) is:
\[ \left(\frac{10 - 4\sqrt{5}}{15}, \frac{10 + 4\sqrt{5}}{15}\right) \]
This corresponds to option (b).