The angle between the hyperbola `xy = c^(2) " and " x^(2) - y^(2) = a^(2) ` is

Options

A. independent of c
B. dependent on a
C. always `pi//3`
D. none of these

Answer

To find the angle between the hyperbola \( xy = c^2 \) and \( x^2 - y^2 = a^2 \), we will follow these steps:

### Step 1: Find the slopes of the tangents to the hyperbolas
1. **For the hyperbola \( xy = c^2 \):**
- Rewrite the equation in terms of \( y \):
\[ y = \frac{c^2}{x} \]
- Differentiate \( y \) with respect to \( x \) to find the slope \( m_1 \):
\[ \frac{dy}{dx} = -\frac{c^2}{x^2} \]
- At the point \( (ct, \frac{c^2}{ct}) \), the slope becomes:
\[ m_1 = -\frac{c^2}{(ct)^2} = -\frac{1}{t^2} \]

2. **For the hyperbola \( x^2 - y^2 = a^2 \):**
- Differentiate the equation implicitly:
\[ 2x - 2y \frac{dy}{dx} = 0 \]
- Rearranging gives:
\[ \frac{dy}{dx} = \frac{x}{y} \]
- At the point \( (ct, \frac{c^2}{ct}) \), we substitute to find \( m_2 \):
\[ m_2 = \frac{ct}{\frac{c^2}{ct}} = t^2 \]

### Step 2: Use the formula for the angle between two curves
The formula for the angle \( \theta \) between two curves with slopes \( m_1 \) and \( m_2 \) is given by:
\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]

### Step 3: Substitute the slopes into the formula
Substituting \( m_1 \) and \( m_2 \):
\[ \tan \theta = \left| \frac{-\frac{1}{t^2} - t^2}{1 + (-\frac{1}{t^2})(t^2)} \right| \]
This simplifies to:
\[ \tan \theta = \left| \frac{-\frac{1 + t^4}{t^2}}{1 - 1} \right| \]
Since the denominator becomes 0, this indicates that \( \tan \theta \) approaches infinity, which means:
\[ \theta = \frac{\pi}{2} \]

### Conclusion
The angle between the hyperbolas \( xy = c^2 \) and \( x^2 - y^2 = a^2 \) is:
\[ \theta = \frac{\pi}{2} \]