Suppose I have f(x) = -3x + 4, along which (3,-5) sits as the center of a circle with a radius of 5. I want to find the point (x2, y2) along the outside of the circle that is the point of tangency for the tangent line that is perpendicular to f(x). Let's call that line g(x). We can solve for (x2,y2) by using the substitution method of solving systems of equations. However, we do not know the equation for g(x). We know that the slope of g(x) is simply -1/slope of f(x). However, the problem is that the y-intercept of g(x) is unknown. We do not technically know any points on this line as of yet.
I'm aware that there are an infinite number of tangent lines for a circle, and because we have not isolated a single point on our line as of yet, we have not isolated any one or set of tangent lines. There must be a way to isolate the tangent line that we want and then go forward and solve for our point of tangency (x2, y2), but I am unaware of an methods.
Any help would be appreciated.