Let the centre be $(h,k)$. Since $k=2h$ and the circle passes through $(0,0)$ and $(8,6)$, we equate squared distances from centre: $h^2+k^2 = (h-8)^2+(k-6)^2$. This simplifies to $16h+12k=100$, or $4h+3k=25$. Substituting $k=2h$, we get $4h+6h=25 \implies 10h=25 \implies h=5/2$. Hence $k=5$. Centre is $(5/2,5)$. Radius squared $r^2 = (5/2)^2+5^2 = 25/4+25 = 125/4$. The equation is $(x-5/2)^2+(y-5)^2 = 125/4$, which simplifies to $x^2+y^2-5x-10y=0$.