Q. 123.6( 27 Votes )

# A point on the hy

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   ………..(1)

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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