Solution:

The first equation can be rewritten as $$y = \frac a2x - \frac c2.$$ Similarly, the second equation can be rewritten as $$y = -\frac 2bx - \frac cb.$$ Since they are perpendicular, we know the slopes multiply to $-1.$

Therefore, $$\frac a2 \cdot \left(-\frac 2b\right) = -1.$$ This means $a=b,$ which implies that $2x+ay=-c.$ We can add this with the first equation to get $$2x+ay+ax-2y=0.$$ Plugging in $(x,y)=(1,-5)$ yields $$2\cdot 1+a-5a-2\cdot (-5)=0.$$ This makes $$2\cdot (6)=4a$$$$a=3.$$

Therefore, $$\begin{align*}c&=3x-2y\\&=3\cdot 1-2\cdot (-5)\\&=13.\end{align*}$$

Thus, the correct answer is E .