Q. A point on the hypotenuse of a right triangle is at distance and from the sides of the triangle. The minimum length of the hypotenuse is

Application of Derivatives Report Error

A

B

C

D

Solution:

Let be a right triangle with hypotenuse such that a point on is at distances and from and respectively, i.e. and
Let . Then,
and
Let be the length of the hypotenuse . Then,

and,
For maximum or minimum, we must have

and

Clearly, for
Thus, is minimum when
The minimum value of is given by

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1. Let
• Column 1 contains information about zeros of and .
• Column 2 contains information about the limiting behavior of and at infinity.
• Column 3 contains information about increasing/decreasing nature of and .

Column 1 Column 2 Column 3

(I) for some (i) (P) is increasing in ()

(II) for some (ii) (Q) is decreasing in ()

(III) for some (ii) (R) is increasing in ()

(IV) for some (ii) (S) is decreasing in ()

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