general form of an equation with center at the origin is

x^2 + y^2 = r^2 where r= the radius.

find the slope of the tangent line, set it equal to the slope of the circle

tangent line slope = 1/2 = dy/dx = y'

2x +2yy' = 0

y' = -x/y = 1/2 for the tangent line y=x/2 +10

plug that into y'

y' =-x/(x/2+10) = 1/2

solve for x =-4, y =x/2+10 = 8

radius = r

r^2 =x^2 + y^2 = 16+64 = 80

the circle's equation is

x^2 +y^2 = 80

before doing any calculations, you know the radius is <10, or 0<r<10

IF the tangent line had slope of 0, the circle would be x^2+y^2 =100

If the tangent line had infinite slope, the circle would degenerate to a dot, with r=0

So the circle has somewhere between 0 and 10 radius.