To attack this problem, one should first write down a formula for this distance. However, to make calculuations much easier, we can just worry about the distance squared and minimize that, as minimizing this will obviously minimize the distance.
D2 = (x - p)2 + (m*x + b - q)2,
where (x, m*x + b) is a point on the line. Now we differentiate this with respect to x and set it equal to zero. Then we solve for x and this yields to
x = (p - m*b + m*q) / (1 + m2).
The rest of the problem is algebra, where we plug in x in the distance formula and simplify the expression. The following is the answer we obtain, submitted by Mr. Andy Young:
