Let, a straight line $L_1$ passing through origin $O$ intersects the lines $10x-8y-10=0 \Rightarrow 5x-4y-5=0 ....\left(L_2\right)$ and $\frac{x}{4}-\frac{y}{5}+1=0 \mathrm{or} 5x-4y+20=0 ...\left(L_3\right)$ at right angles and at the points $P$ and $Q$ respectively. 

We can see $L_2$ and $L_3$ are parallel lines.

So, perpendicular distance from origin to $L_2=OP=\frac{\left|5\left(0\right)-4\left(0\right)-5\right|}{\sqrt{5^2+4^2}}=\frac{5}{\sqrt{41}}$

and 

$L_3=OQ=\frac{\left|5\left(0\right)-4\left(0\right)+20\right|}{\sqrt{5^2+4^2}}=\frac{20}{\sqrt{41}}$

Hence, $OP:OQ = 1:4$