The angle between the lines `x^(2)+4xy+y^(2)=0` is

Options

A. `60^(@)`
B. `15%^(@)`
C. `30^(@)`
D. `45^(@)`

Explanation

To find the angle between the lines represented by the equation \(x^2 + 4xy + y^2 = 0\), we can follow these steps:

### Step 1: Identify the coefficients
The equation of the pair of straight lines is in the form \(ax^2 + 2hxy + by^2 = 0\). We need to identify the coefficients \(a\), \(b\), and \(h\) from the given equation.

From the equation \(x^2 + 4xy + y^2 = 0\):
- \(a = 1\) (coefficient of \(x^2\))
- \(b = 1\) (coefficient of \(y^2\))
- \(2h = 4\) (coefficient of \(xy\)), thus \(h = 2\).

### Step 2: Use the angle formula
The formula to find the angle \(\theta\) between the two lines is given by:
\[ \tan \theta = \frac{2\sqrt{h^2 - ab}}{a + b} \]

### Step 3: Substitute the values into the formula
Now, substituting the values we found:
- \(a = 1\)
- \(b = 1\)
- \(h = 2\)

We can calculate:
\[ \tan \theta = \frac{2\sqrt{2^2 - (1)(1)}}{1 + 1} \]
\[ = \frac{2\sqrt{4 - 1}}{2} \]
\[ = \frac{2\sqrt{3}}{2} \]
\[ = \sqrt{3} \]

### Step 4: Find the angle \(\theta\)
To find \(\theta\), we take the inverse tangent:
\[ \theta = \tan^{-1}(\sqrt{3}) \]

From trigonometric values, we know:
\[ \tan(60^\circ) = \sqrt{3} \]
Thus,
\[ \theta = 60^\circ \]

### Conclusion
The angle between the lines represented by the equation \(x^2 + 4xy + y^2 = 0\) is \(60^\circ\).

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