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 find the value of \( k \) such that the equation \( 4x^2 + ky^2 = 18 \) represents a rectangular hyperbola, we can follow these steps:

### Step 1: Rewrite the equation in standard form
We start with the given equation:
\[ 4x^2 + ky^2 = 18 \]
To rewrite it in standard form, we divide the entire equation by 18:
\[ \frac{4x^2}{18} + \frac{ky^2}{18} = 1 \]
This simplifies to:
\[ \frac{x^2}{\frac{18}{4}} + \frac{y^2}{\frac{18}{k}} = 1 \]
Which can be further simplified to:
\[ \frac{x^2}{\frac{9}{2}} + \frac{y^2}{\frac{18}{k}} = 1 \]

### Step 2: Identify the conditions for a rectangular hyperbola
For the hyperbola to be rectangular, we need the relationship between \( a^2 \) and \( b^2 \) to satisfy:
\[ b^2 = a^2 \]
where \( a^2 = \frac{9}{2} \) and \( b^2 = \frac{18}{k} \).

### Step 3: Set up the equation
Using the condition for a rectangular hyperbola:
\[ \frac{18}{k} = \frac{9}{2} \]

### Step 4: Solve for \( k \)
Cross-multiplying gives:
\[ 18 \cdot 2 = 9k \]
This simplifies to:
\[ 36 = 9k \]
Dividing both sides by 9:
\[ k = \frac{36}{9} = 4 \]

### Step 5: Determine the sign of \( k \)
Since we are dealing with a rectangular hyperbola, we need to ensure that the equation maintains the hyperbolic form. The equation must have a negative sign in front of the \( y^2 \) term. This means \( k \) must be negative. Thus, we take:
\[ k = -4 \]

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