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

Options

A. `0^(@)`
B. `45^(@)`
C. `90^(@)`
D. `180^(@)`

Answer

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

### Step 1: Rewrite the equation in standard form
The given equation is:
\[ x^2 - y^2 = 0 \]
We can rewrite this as:
\[ x^2 - y^2 + 0 \cdot xy = 0 \]
This helps us identify the coefficients for the standard form of the conic section.

### Step 2: Identify coefficients
From the equation \( Ax^2 + By^2 + 2Hxy = 0 \), we can identify:
- \( A = 1 \)
- \( B = -1 \)
- \( H = 0 \)

### Step 3: Use the formula for the angle between two lines
The formula for the angle \( \theta \) between two lines represented by the equation is given by:
\[ \tan \theta = \left| \frac{2\sqrt{H^2 - AB}}{A + B} \right| \]
Substituting the values of \( A \), \( B \), and \( H \):
\[ \tan \theta = \left| \frac{2\sqrt{0^2 - (1)(-1)}}{1 + (-1)} \right| \]

### Step 4: Simplify the expression
Calculating the values:
\[ \tan \theta = \left| \frac{2\sqrt{0 + 1}}{1 - 1} \right| = \left| \frac{2\sqrt{1}}{0} \right| = \left| \frac{2}{0} \right| \]
Since division by zero is undefined, \( \tan \theta \) approaches infinity.

### Step 5: Determine the angle
Since \( \tan \theta \) is infinite, this implies:
\[ \theta = 90^\circ \]

### Conclusion
Thus, the angle between the lines represented by the equation \( x^2 - y^2 = 0 \) is:
\[ \boxed{90^\circ} \]