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

Options

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

Answer

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

### Step 1: Rewrite the equation
The given equation is:
\[ x^2 - y^2 - 2y - 1 = 0 \]
We can rearrange this equation to identify it as a pair of straight lines.

### Step 2: Identify coefficients
The general form of the equation of a pair of straight lines is:
\[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \]
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)
- \(g = 0\) (since there is no \(x\) term)
- \(f = -1\) (coefficient of \(y\))
- \(c = -1\)

### Step 3: Use the formula for the angle between the lines
The formula for the angle \(\theta\) between the two lines represented by the equation is given by:
\[ \tan \theta = \frac{2\sqrt{h^2 - ab}}{a + b} \]
Substituting the values we found:
- \(a = 1\)
- \(b = -1\)
- \(h = 0\)

### Step 4: Calculate \(a + b\)
First, calculate \(a + b\):
\[ a + b = 1 - 1 = 0 \]

### Step 5: Calculate \(h^2 - ab\)
Next, calculate \(h^2 - ab\):
\[ h^2 - ab = 0^2 - (1)(-1) = 0 + 1 = 1 \]

### Step 6: Substitute into the angle formula
Now substitute these values into the angle formula:
\[ \tan \theta = \frac{2\sqrt{1}}{0} \]
This results in:
\[ \tan \theta = \frac{2}{0} \]
Since division by zero is undefined, this indicates that \(\tan \theta\) approaches infinity.

### Step 7: Determine the angle
The tangent of an angle is infinite when the angle is \(90^\circ\). Therefore, we conclude that:
\[ \theta = 90^\circ \]

### Final Answer
The angle between the straight lines is \(90^\circ\).

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