To find the constant of variation \( k \) for the direct variation given by the equation \( 2x + 6y = 0 \), we can first rearrange this equation into a form that highlights the relationship between \( x \) and \( y \).
Start by isolating \( y \):
\[ 6y = -2x \]
Now, divide both sides by 6:
\[ y = -\frac{2}{6}x \]
Simplifying \(-\frac{2}{6}\):
\[ y = -\frac{1}{3}x \]
In the context of direct variation, this can be expressed in the form \( y = kx \), where \( k \) is the constant of variation. Here, we can see that:
\[ k = -\frac{1}{3} \]
Thus, the constant of variation \( k \) is:
\[ \boxed{-\frac{1}{3}} \]