The equations (s) to common tangent (s) to the two hyperbola `x^(2)/a^(2) - y^(2)/b^(2) = 1 " and " y^(2)/a^(2) - x^(2)/b^(2) = 1` is /are

Options

A. `y = x + sqrt(a^(2) - b^(2))`
B. `y = x - sqrt(a^(2) - b^(2))`
C. `y = - x + sqrt(a^(2) - b^(2))`
D. `y = - x - sqrt(a^(2) - b^(2))`

Answer

To find the equations of the common tangents to the hyperbolas given by:

1. \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)
2. \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \)

we will follow these steps:

### Step 1: Write the equations of the tangents

For the first hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the equation of the tangent can be expressed as:

\[ y = mx \pm \sqrt{a^2m^2 - b^2} \]

For the second hyperbola \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \), the equation of the tangent is:

\[ y = mx \pm \sqrt{a^2 - b^2m^2} \]

### Step 2: Set the constants equal

Since we are looking for common tangents, we set the constants from both tangent equations equal to each other:

\[ \sqrt{a^2m^2 - b^2} = \sqrt{a^2 - b^2m^2} \]

### Step 3: Square both sides

Squaring both sides to eliminate the square roots gives us:

\[ a^2m^2 - b^2 = a^2 - b^2m^2 \]

### Step 4: Rearrange the equation

Rearranging the equation leads to:

\[ a^2m^2 + b^2m^2 = a^2 + b^2 \]

This can be factored as:

\[ m^2(a^2 + b^2) = a^2 + b^2 \]

### Step 5: Solve for \( m^2 \)

Assuming \( a^2 + b^2 \neq 0 \), we can divide both sides by \( a^2 + b^2 \):

\[ m^2 = 1 \]

### Step 6: Find the values of \( m \)

Taking the square root gives us the slopes of the common tangents:

\[ m = \pm 1 \]

### Step 7: Substitute the values of \( m \) back into the tangent equations

1. For \( m = 1 \):

\[ y = x \pm \sqrt{a^2 - b^2} \]

2. For \( m = -1 \):

\[ y = -x \pm \sqrt{a^2 - b^2} \]

### Final Equations of the Common Tangents

Thus, the equations of the common tangents are:

1. \( y = x + \sqrt{a^2 - b^2} \)
2. \( y = x - \sqrt{a^2 - b^2} \)
3. \( y = -x + \sqrt{a^2 - b^2} \)
4. \( y = -x - \sqrt{a^2 - b^2} \)

### Summary of Results

All four equations represent the common tangents to the given hyperbolas.

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