The angle between the pair of lines given by the equation `x^(2)+2xy-y^(2)=0` is

Options

A. `pi//3`
B. `pi//6`
C. `pi//2`
D. `0`

Answer

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

### Step 1: Identify the coefficients
The given equation can be compared with the general form of the equation of a pair of straight lines:
\[ ax^2 + 2hxy + by^2 = 0 \]
From the equation \( x^2 + 2xy - y^2 = 0 \), we can identify the coefficients:
- \( a = 1 \)
- \( b = -1 \)
- \( h = 1 \)

### Step 2: Calculate the discriminant
To determine the angle between the lines, we need to check if the lines are perpendicular. The condition for the lines to be perpendicular is given by:
\[ h^2 - ab = 0 \]
Substituting the values we found:
\[ h^2 - ab = 1^2 - (1)(-1) = 1 - (-1) = 1 + 1 = 2 \]
Since \( h^2 - ab \neq 0 \), we need to calculate the angle.

### Step 3: Calculate the angle between the lines
The angle \( \theta \) between the two lines can be calculated using the formula:
\[ \tan \theta = \frac{2h}{a - b} \]
Substituting the values:
\[ \tan \theta = \frac{2 \cdot 1}{1 - (-1)} = \frac{2}{1 + 1} = \frac{2}{2} = 1 \]

### Step 4: Find the angle
Now, we find \( \theta \):
\[ \theta = \tan^{-1}(1) = \frac{\pi}{4} \]
This means the lines are at an angle of \( \frac{\pi}{4} \) radians.

### Step 5: Determine if the lines are perpendicular
Since we calculated \( h^2 - ab = 2 \) and not 0, we conclude that the lines are not perpendicular. However, since we have \( \tan \theta = 1 \), we can also conclude that the angle between the lines is \( \frac{\pi}{4} \) radians.

### Final Conclusion
Thus, the angle between the pair of lines given by the equation \( x^2 + 2xy - y^2 = 0 \) is \( \frac{\pi}{4} \) radians.

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