In diophantine $3b^2=a^2$ where $a$ and $b$ are coprime, does $3|a$? Integers $a$ and $b$ are co-prime and $3\cdot b^2=a^2$.
$3\cdot b^2=a^2$, implies $a^2$ is divisible by 3 since, $3b^2$ is divisible by 3.
Is $a$ divisible by 3?

Yep; in general, if $p|ab$ then $p|a$ or $p|b$ ($p$ is a prime). In your case, $3|a^2$ so $3|a$ or $3|a$; thus $a$ is divisible by $3$
