A point on the hypotenuse of a right angled triangle is at distances 'a' and 'b' from the sides .

Show that the length of the hypotenuse is at least

{

__raised to 2/3 +__**a****raised to 2/3}the whole raised to 3/2**__b__

Let AOB be a right triangle with hypotenuse AB such that a point P on AB is distance *a* and *b* from OA and OB respectively. i.e. PL =* a* and PM = *b*

Let ∠OAB = θ Then,

AP = *a* cosec θ and BP = *b* sec θ

Let *l* be the length of the hypotenuse AB. Then

*l* = AP + BP

⇒* l* = *a* cosec θ + *b* sec θ

For maximum or minimum, we must have

Clearly,

Thus,* l *is minimum when

The minimum value of *l *is given by

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