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
{ a raised to 2/3 + b raised to 2/3}the whole raised to 3/2


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


Thus, l is minimum when

The minimum value of l is given by

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