Q. 124.1( 15 Votes )
A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle.
Show that the maximum length of the hypotenuse is 
Answer :
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,
Ac = 
Now, PC = b cosecθ
And AP = a secθ
⇒ AC = AP + PC
⇒ AC = a secθ + b cosecθ
Now, if 
⇒ asecθ tanθ = bcosecθ cotθ
⇒ 
⇒ asin3θ =bcos3θ
⇒ 
⇒ 
………..(1)
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,
Ac = 
Therefore, the maximum length of the hypotenuses is 
.
Rate this question :
How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
PREVIOUSA window is in the form of a rectangle surmounted by a semi - circular opening.The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.NEXTFind the points at which the function f given by f (x) = (x – 2)4 (x + 1)3 has(i) local maxima (ii) local minima(iii) point of inflexion
RELATED QUESTIONS :
Show that the maximum value of 
is 
Mark (√) against the correct answer in the following:
At 
is
RS Aggarwal - Mathematics
Find the point of local maxima or local minima or local minima and the corresponding local maximum and minimum values of each of the following functions:
RS Aggarwal - Mathematics
Find the point of local maxima or local minima or local minima and the corresponding local maximum and minimum values of each of the following functions:
RS Aggarwal - Mathematics
