Closest Point

The shortest distance of $(a, b)$ Point from the line, $x - y - 8 = 0$ is, $|\frac{a - b - 7} {\sqrt{2}}|$ As the point $(a, b)$ lies on the parabola, $y = x^{2} - 9x + 25$ , $a = x$ $b = x^{2} - 9x + 25$ Hence the shortest distance becomes, after substituting the values and refactoring: $|\frac{(x-5)^{2} + 8}{\sqrt{2}}|$ The minimum value is achieved, when $x = 5$ . Hence, $a = 5, b = 5; a + b = 10$ .

Also can solve it use calculus (derivates)