There are several different ways to do that.
1) Solve the equation of the line for y and replace y in the equation of the circle by that expression,, getting a quadratic equation in x. The line is tangent to the circle if and only if the quadratic equation has a double root- if it reduces to .
2) By completing the square, find the center of the circle and so find the equation, and then slope of the line from the center of the circle to (2, 6), a radius of the circle. Show that (2, 6) satisfies the equations of both line and circle and show that the line is perpendicular to the radius by showing that the product of their slopes is -1.
3) Show that x= 2, y= 6 satisfies the equations of both line and circle and show that the slope of the line is equal to the derivative of the equation of the circle at that point.