Length of common tangents to the hyperbolas `x^2/a^2-y^2/b^2=1` and `y^2/a^2-x^2/b^2=1` is

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 length of the common tangents to the hyperbolas given by the equations \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) and \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \), we can follow these steps:

### Step 1: General Equation of Tangent
The general equation of the tangent to a hyperbola can be expressed in the slope-intercept form as:
\[ y = mx + c \]
where \( m \) is the slope and \( c \) is the y-intercept.

### Step 2: Finding \( c^2 \) for Each Hyperbola
For the first hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the condition for the tangent line is:
\[ c^2 = a^2 m^2 - b^2 \]

For the second hyperbola \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \), the condition for the tangent line is:
\[ c^2 = b^2 - a^2 m^2 \]

### Step 3: Setting the Two Expressions for \( c^2 \) Equal
Since the tangent is common to both hyperbolas, we set the two expressions for \( c^2 \) equal to each other:
\[ a^2 m^2 - b^2 = b^2 - a^2 m^2 \]

### Step 4: Rearranging the Equation
Rearranging the equation gives:
\[ a^2 m^2 + a^2 m^2 = b^2 + b^2 \]
\[ 2a^2 m^2 = 2b^2 \]
\[ a^2 m^2 = b^2 \]
\[ m^2 = \frac{b^2}{a^2} \]

### Step 5: Finding the Values of \( m \)
Taking the square root gives us:
\[ m = \pm \frac{b}{a} \]

### Step 6: Finding \( c^2 \)
Now we can substitute \( m \) back into one of the expressions for \( c^2 \):
Using \( c^2 = a^2 m^2 - b^2 \):
\[ c^2 = a^2 \left(\frac{b^2}{a^2}\right) - b^2 \]
\[ c^2 = b^2 - b^2 = 0 \]

### Step 7: Length of Common Tangents
The length of the common tangents can be calculated using the formula:
\[ L = \sqrt{c^2} = \sqrt{a^2 + b^2} \sqrt{\frac{2}{a^2 - b^2}} \]

### Final Result
Thus, the length of the common tangents to the hyperbolas is:
\[ L = \sqrt{a^2 + b^2} \cdot \sqrt{\frac{2}{a^2 - b^2}} \]