Use the fact that the tangent is perpendicular to the radius. Suppose that the circle has centre (a,b) and radius r. If the tangent has gradient 1 then the radius will have gradient –1, so it will be in the direction of one of the vectors

. So the point where the tangent touches the circle will be

. The equation of the line with gradient 1 through that point is then

. That gives you the equations of the two tangents having gradient 1.

Notice that if a, b and r are whole numbers then the equations of the tangents will also have whole numbers as their constant terms.