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 the equations:

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

we can follow these steps:

### Step 1: Write the equations of the tangents to both hyperbolas

For the first hyperbola, the equation of the tangent in slope-intercept form is given by:

\[ y = mx + c_1 \]

Where \( c_1 = \sqrt{b^2 m^2 + a^2} \).

For the second hyperbola, the equation of the tangent is:

\[ y = mx + c_2 \]

Where \( c_2 = \sqrt{a^2 m^2 + b^2} \).

### Step 2: Set the constants equal

Since we are looking for common tangents, we set \( c_1 = c_2 \):

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

### Step 3: Square both sides to eliminate the square roots

Squaring both sides gives:

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

### Step 4: Rearranging the equation

Rearranging this equation, we get:

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

Factoring out \( m^2 \):

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

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

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

\[ m^2 = 1 \]

Thus, \( m = \pm 1 \).

### Step 6: Write the equations of the tangents

Using the slopes \( m = 1 \) and \( m = -1 \), we can write the equations of the tangents:

1. For \( m = 1 \):

\[ y = x + c \]

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

\[ y = -x + c \]

### Step 7: Find the value of \( c \)

To find \( c \), we substitute back into either hyperbola's tangent equation. For instance, using \( m = 1 \):

\[ c = \sqrt{b^2(1^2) + a^2} = \sqrt{b^2 + a^2} \]

Thus, the equations of the common tangents are:

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

### Final Answer

The equations of the common tangents to the two hyperbolas are:

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

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