If the circles $x^2+y^2+2ax+cy+a=0$ and 

$x^2+y^2-3ax+dy-1=0$ intersect in two distinct points$P$ and $Q$, then the line $5x + by - a = 0$ passes through $P$ and $Q$, for

 

The equation of the common chord of the given circles is

$5ax+(c-d)y+(a+1)=0\dots \left(\mathrm{i}\right)\left[\text{ Using: }{S}_1-S_2=0\right]$

But, the equation of the common chord is given as

$5x+by-a=0$

Comparing (i) and (ii), we get

$\begin{array}{l}\frac{5a}{5}=\frac{c-d}{b}=\frac{a+1}{-a}\\ \Rightarrow \frac{a}{1}=\frac{c-d}{b}=-\frac{a+1}{a}\Rightarrow a=-\frac{(a+1)}{a}\Rightarrow a^2+a+1=0\end{array}$

Clearly, $a^2+a+1=0$ has no real roots. So, there is no real value of a for which $5x + by - a= 0$ passes through $P$ and $Q.$