Given equation is
$8 x^2+8 x y+2 y^2+26 x+13 y+15=0$ ...(i)
On writing Eq. (i) as quadratic equation in $x$, we get
$8 x^2+2 x(4 y+13)+2 y^2+13 y+15=0$
$x=\frac{-2(4 y+13) \pm \sqrt{4(4 y+13)^2-32\left(2 y^2+13 y+15\right.}}{16}$
$\Rightarrow \quad x=\frac{-(4 y+13) \pm 7}{8}$
$\begin{aligned} & \Rightarrow \quad 8 x=-4 y-13+7 \text { or } 8 x=-4 y-13-7 \\ & \Rightarrow \quad 4 x+2 y+3=0 \text { or } 2 x+y+5=0 \\ & \Rightarrow \quad 2 x+y+3 / 2=0 \text { or } 2 x+y+5=0\end{aligned}$
$\therefore \quad d=\left|\frac{c_1-c_2}{\sqrt{a^2+b^2}}\right|$,
where, $c_1=5, c_2=3 / 2, a=2, b=1$
$d=\left|\frac{5-3 / 2}{\sqrt{2^2+1^2}}\right|=\frac{7}{2 \sqrt{5}}$