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 hyperbolas given by the equations \(xy = c^2\) and \(x^2 - y^2 = a^2\), we will follow these steps:

### Step 1: Find the slopes of the tangents to the hyperbolas at their intersection points.

1. **Hyperbola 1**: The equation is \(xy = c^2\).
- We can express \(y\) in terms of \(x\):
\[ y = \frac{c^2}{x} \]
- To find the slope, we differentiate \(y\) with respect to \(x\):
\[ \frac{dy}{dx} = -\frac{c^2}{x^2} \]
- Let’s denote this slope as \(m_1 = -\frac{c^2}{x^2}\).

2. **Hyperbola 2**: The equation is \(x^2 - y^2 = a^2\).
- We can express \(y\) in terms of \(x\):
\[ y = \sqrt{x^2 - a^2} \]
- Differentiating this implicitly gives:
\[ 2x - 2y\frac{dy}{dx} = 0 \implies \frac{dy}{dx} = \frac{x}{y} \]
- Let’s denote this slope as \(m_2 = \frac{x}{y}\).

### Step 2: Find the angle between the two tangents.

The angle \(\theta\) between two curves at their intersection point can be found using the formula:
\[ \tan \theta = \frac{m_1 - m_2}{1 + m_1 m_2} \]

### Step 3: Substitute the slopes into the formula.

Substituting \(m_1\) and \(m_2\) into the angle formula:
\[ \tan \theta = \frac{-\frac{c^2}{x^2} - \frac{x}{y}}{1 + \left(-\frac{c^2}{x^2}\right)\left(\frac{x}{y}\right)} \]

### Step 4: Simplify the expression.

1. **Numerator**:
\[ -\frac{c^2}{x^2} - \frac{x}{y} = -\frac{c^2}{x^2} - \frac{x^2}{xy} \]
Combining the fractions gives:
\[ = -\frac{c^2y + x^3}{x^2y} \]

2. **Denominator**:
\[ 1 - \frac{c^2x}{x^2y} = 1 - \frac{c^2}{xy} \]

### Step 5: Analyze the result.

As we analyze the slopes, we find that:
- If \(m_1 m_2 = -1\), then \(\tan \theta\) becomes undefined, which implies that \(\theta = \frac{\pi}{2}\).

### Conclusion:

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