Find the center coordinates and radius r from the standard equation of a circle. When a straight line is tangent to a circle, the distance from the center of the circle to the straight line is equal to the radius of the circle. List relationships using the distance formula from point to straight line. After sorting out, the basic inequality is used to deform, let m+n=x, and the inequality about x is obtained, and the solution set of the inequality is obtained to get the value range of x, that is, the value range of m+n.

This topic examines the positional relationship between a straight line and a circle, involving knowledge such as the distance formula from a point to a straight line, the basic inequality, and the solution of a quadratic inequality. Using the idea of transformation and substitution, when a straight line is tangent to a circle, the distance from the center of the circle to the straight line is equal to the radius of the circle. Mastering this property skillfully is the key to solve this problem.

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