The angle between the straight lines `x^(2)-y^(2)-2x-1=0`, is

Options

A. `90^(@)`
B. `60^(@)`
C. `75^(@)`
D. `36^(@)`

Answer

To find the angle between the straight lines represented by the equation \(x^2 - y^2 - 2x - 1 = 0\), we can follow these steps:

### Step 1: Rewrite the equation
The given equation is:
\[ x^2 - y^2 - 2x - 1 = 0 \]
We can rearrange this equation to match the standard form of a conic section.

### Step 2: Identify coefficients
The standard form of a conic section is:
\[ ax^2 + by^2 + 2hxy + gx + 2fy + c = 0 \]
From our equation, we can identify:
- \(a = 1\) (coefficient of \(x^2\))
- \(b = -1\) (coefficient of \(y^2\))
- \(h = 0\) (since there is no \(xy\) term)
- \(g = -2\) (coefficient of \(x\))
- \(f = 0\) (coefficient of \(y\))
- \(c = -1\) (constant term)

### Step 3: Use the formula for the angle between the lines
The angle \(\theta\) between the two lines can be found using the formula:
\[ \tan \theta = \frac{|\sqrt{(h^2 - ab)}|}{a + b} \]
Substituting the values we have:
- \(h = 0\)
- \(a = 1\)
- \(b = -1\)

### Step 4: Calculate \(h^2 - ab\)
Now, we calculate:
\[ h^2 - ab = 0^2 - (1)(-1) = 0 + 1 = 1 \]
Thus, \(\sqrt{h^2 - ab} = \sqrt{1} = 1\).

### Step 5: Calculate \(a + b\)
Next, we calculate:
\[ a + b = 1 - 1 = 0 \]

### Step 6: Substitute into the formula
Now substituting these values into the formula for \(\tan \theta\):
\[ \tan \theta = \frac{2 \cdot 1}{0} \]
Since division by zero is undefined, this implies that \(\tan \theta\) approaches infinity.

### Step 7: Determine the angle
When \(\tan \theta\) is infinite, it means that:
\[ \theta = 90^\circ \]

### Conclusion
Thus, the angle between the straight lines is:
\[ \theta = 90^\circ \]