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.