If the equation `4x^(2) + ky^(2) = 18` represents a rectangular hyperbola, then k =

Options

A. 4
B. `-4`
C. 3
D. none of these

Answer

To determine the value of \( k \) for the equation \( 4x^2 + ky^2 = 18 \) to represent a rectangular hyperbola, we can follow these steps:

### Step 1: Rewrite the equation in standard form
We start with the given equation:
\[ 4x^2 + ky^2 = 18 \]
Dividing the entire equation by 18 gives:
\[ \frac{4x^2}{18} + \frac{ky^2}{18} = 1 \]
This simplifies to:
\[ \frac{2x^2}{9} + \frac{ky^2}{18} = 1 \]

### Step 2: Identify the coefficients
From the equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), we can identify:
\[ a^2 = \frac{9}{2} \quad \text{and} \quad -b^2 = -\frac{18}{k} \]
For the hyperbola to be rectangular, we need the condition \( a^2 = b^2 \).

### Step 3: Set the condition for a rectangular hyperbola
Setting \( a^2 = b^2 \):
\[ \frac{9}{2} = \frac{18}{k} \]

### Step 4: Solve for \( k \)
Cross-multiplying gives:
\[ 9k = 36 \]
Thus,
\[ k = \frac{36}{9} = 4 \]

### Step 5: Determine the sign of \( k \)
Since we are looking for a rectangular hyperbola, we need \( k \) to be negative. Therefore, we take:
\[ k = -4 \]

### Conclusion
The value of \( k \) for which the equation \( 4x^2 + ky^2 = 18 \) represents a rectangular hyperbola is:
\[ \boxed{-4} \]