The equation of a locus is $y^{2}+2ax+2by+c=0$. Then

Given equation of locus is $\displaystyle { y }^{ 2 }+2ax+2by+c=0$, which can be written as $\displaystyle { (y+b) }^{ 2 }=-4\left(\frac { a }{ 2 } \right)\left(x+\frac { c-{ b }^{ 2 } }{ 2a } \right)$ which is locus of parabola.

For parabola with equation $\displaystyle {y}^{2}=4ax$, $\displaystyle 4a$ is length of latus rectum.

Then for given locus, length of latus rectum is $\displaystyle 4\left(\frac{a}{2}\right)=2a$.