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

Options

A. `30^(@)`
B. `45^(@)`
C. `60^(@)`
D. `90^(@)`

Answer

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

### Step 1: Identify coefficients
The given equation is in the form \(ax^2 + 2hxy + by^2 = 0\). Here, we can identify:
- \(a = 1\)
- \(b = 1\)
- \(h = 2\)

### Step 2: Use the formula for the angle between two lines
The formula for the angle \(\theta\) between the two lines represented by the equation 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 of \(a\), \(b\), and \(h\):
\[ \tan \theta = \frac{2\sqrt{2^2 - (1)(1)}}{1 + 1} \]
Calculating inside the square root:
\[ \tan \theta = \frac{2\sqrt{4 - 1}}{2} = \frac{2\sqrt{3}}{2} = \sqrt{3} \]

### Step 4: Find the angles
Since \(\tan \theta = \sqrt{3}\), we can find the angles:
\[ \theta = \frac{\pi}{3} \quad \text{or} \quad \theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \]

### Conclusion
Thus, the angles between the straight lines are \(\frac{\pi}{3}\) and \(\frac{2\pi}{3}\).

### Final Answer
The angle between the straight lines is \(\frac{\pi}{3}\) and \(\frac{2\pi}{3}\).

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