Answer :

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 .

Now, we have,

Ac =

Now, PC = b cosecθ

And AP = a secθ

AC = AP + PC

AC = a secθ + b cosecθ

Now, if

asecθ tanθ = bcosecθ cotθ

asin3θ =bcos3θ


So, it is clear that < 0 when

Therefore, by second derivative test, the length of the hypotenuse is the maximum when

Now, when , we get,

Ac =

Therefore, the maximum length of the hypotenuses is .

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