The value `k` for which `4x^2+8x y+k y^2=9` is the equation of a pair of straight lines is__________

Explanation

To find the value of \( k \) for which the equation \( 4x^2 + 8xy + ky^2 = 9 \) represents a pair of straight lines, we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 4x^2 + 8xy + ky^2 - 9 = 0 \]
This can be compared to the general form of a conic section:
\[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \]

### Step 2: Identify coefficients
From the equation \( 4x^2 + 8xy + ky^2 - 9 = 0 \), we can identify the coefficients:
- \( a = 4 \)
- \( h = 4 \) (since \( 2h = 8 \))
- \( b = k \)
- \( g = 0 \)
- \( f = 0 \)
- \( c = -9 \)

### Step 3: Use the condition for a pair of straight lines
For the equation to represent a pair of straight lines, the determinant \( \Delta \) must be equal to zero. The determinant is given by:
\[ \Delta = abc + 2fgh - af^2 - bg^2 - ch^2 \]
Substituting the values we have:
\[ \Delta = (4)(k)(-9) + 2(0)(4)(0) - (4)(0)^2 - (k)(0)^2 - (-9)(4^2) \]
This simplifies to:
\[ \Delta = -36k + 0 - 0 - 0 + 144 \]
So, we have:
\[ \Delta = -36k + 144 \]

### Step 4: Set the determinant to zero
To find the value of \( k \), we set \( \Delta = 0 \):
\[ -36k + 144 = 0 \]

### Step 5: Solve for \( k \)
Now, we solve for \( k \):
\[ -36k = -144 \]
\[ k = \frac{144}{36} = 4 \]

### Final Answer
Thus, the value of \( k \) for which \( 4x^2 + 8xy + ky^2 = 9 \) represents a pair of straight lines is:
\[ \boxed{4} \]