Question:

For a focal chord of a parabola $y^2 = 4ax$, if $P$ and $Q$ are the endpoints with focal distances $SP = l_1$ and $SQ = l_2$, then using the focal chord property: $\frac{1}{l_1} + \frac{1}{l_2} = \frac{1}{a}$. For a focal chord, if $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$ are endpoints, then $t_1t_2 = -1$. The focal distances are $SP = a(1+t_1^2)$ and $SQ = a(1+t_2^2)$. Therefore: $\frac{1}{SP} + \frac{1}{SQ} = \frac{1}{a(1+t_1^2)} + \frac{1}{a(1+t_2^2)} = \frac{(1+t_2^2)+(1+t_1^2)}{a(1+t_1^2)(1+t_2^2)}$. Since $t_1t_2 = -1$, we get $\frac{2+(t_1^2+t_2^2)}{a(1+t_1^2+t_2^2+t_1^2t_2^2)} = \frac{2}{a}$. The specific numerical answer depends on the given parabola equation (likely $a = \frac{5}{\sqrt{13}}$ to yield $\frac{2\sqrt{13}}{5}$).