If the lines represented by `3y^(2)+9xy+kx^(2)=0` are perpendicular to eachother then `k=`

Options

A. 1
B. -2
C. -3
D. 0

Explanation

To determine the value of \( k \) for which the lines represented by the equation \( 3y^2 + 9xy + kx^2 = 0 \) are perpendicular to each other, we can follow these steps:

### Step 1: Identify the coefficients
The given equation can be rewritten in the standard form of a pair of straight lines:
\[ kx^2 + 9xy + 3y^2 = 0 \]
Here, we can identify:
- \( A = k \)
- \( B = 9 \)
- \( C = 3 \)

### Step 2: Use the condition for perpendicular lines
For two lines represented by the equation \( Ax^2 + 2Hxy + By^2 = 0 \) to be perpendicular, the condition is:
\[ H^2 = AB \]
In our case, we have:
- \( H = \frac{9}{2} \) (since \( 2H = 9 \))
- \( A = k \)
- \( B = 3 \)

### Step 3: Substitute the values into the condition
Now, substituting the values into the perpendicularity condition:
\[ \left(\frac{9}{2}\right)^2 = k \cdot 3 \]
Calculating \( \left(\frac{9}{2}\right)^2 \):
\[ \frac{81}{4} = k \cdot 3 \]

### Step 4: Solve for \( k \)
To isolate \( k \), we can rearrange the equation:
\[ k = \frac{81}{4 \cdot 3} = \frac{81}{12} = \frac{27}{4} \]

### Step 5: Conclusion
Thus, the value of \( k \) for which the lines are perpendicular is:
\[ \boxed{\frac{27}{4}} \]