If the distance between two directrices of a rectangular hyperbola is 15, then the distance between its foci in units is:

Options

A. `15sqrt(2)`
B. 30
C. 60
D. 45

Answer

To solve the problem step by step, we need to find the distance between the foci of a rectangular hyperbola given that the distance between its directrices is 15 units.

### Step 1: Understand the properties of a rectangular hyperbola
For a rectangular hyperbola, the relationship between the semi-major axis \(a\) and the eccentricity \(e\) is given by:
- The directrices are located at \(x = \pm \frac{a}{e}\).

### Step 2: Calculate the distance between the directrices
The distance between the two directrices is given by:
\[ \text{Distance between directrices} = 2 \cdot \frac{a}{e} \]
According to the problem, this distance is 15 units:
\[ 2 \cdot \frac{a}{e} = 15 \]

### Step 3: Solve for \(a\) in terms of \(e\)
From the equation above, we can express \(2a\) in terms of \(e\):
\[ \frac{a}{e} = \frac{15}{2} \]
Multiplying both sides by \(e\):
\[ a = \frac{15}{2} e \]

### Step 4: Find the distance between the foci
The foci of a rectangular hyperbola are located at \((ae, 0)\) and \((-ae, 0)\). Therefore, the distance between the foci is given by:
\[ \text{Distance between foci} = 2ae \]

### Step 5: Substitute the value of \(a\)
Substituting \(a = \frac{15}{2} e\) into the distance formula:
\[ \text{Distance between foci} = 2 \cdot \left(\frac{15}{2} e\right) \cdot e = 15e^2 \]

### Step 6: Determine the value of \(e\) for a rectangular hyperbola
For a rectangular hyperbola, the eccentricity \(e\) is given by:
\[ e = \sqrt{2} \]

### Step 7: Substitute \(e\) into the distance formula
Now substituting \(e = \sqrt{2}\) into the distance between foci:
\[ \text{Distance between foci} = 15 \cdot (\sqrt{2})^2 = 15 \cdot 2 = 30 \]

### Conclusion
Thus, the distance between the foci of the rectangular hyperbola is:
\[ \boxed{30} \text{ units} \]