Q. 123.6( 27 Votes )
A point on the hy
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,
Now, PC = b cosecθ
And AP = a secθ
⇒ AC = AP + PC
⇒ AC = a secθ + b cosecθ
⇒ 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,
Therefore, the maximum length of the hypotenuses is .
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