A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle.

Show that the minimum length of the hypotenuse is

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Solution

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 θ

And, 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 when

Now, when, we have:

Hence, the maximum length of the hypotenuses is.

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