Let ΔABC be right-angled at B. Let AB = x and BC = y.Let P be a point on the hypotenuse of the triangle such that P is at a distance of a and b from the sides AB and BC respectively.Let ∠C = θ.We have,Now,PC = b cosec θAnd, AP = a sec θ∴AC = AP + PC⇒ AC = b cosec θ + a sec θ … (1)Therefore, by second derivative test, the length of the hypotenuse is the maximum whenNow, when, we have:Hence, the maximum length of the hypotenuses is.
