Coordinate geometry doubt

Question

OABC is a rectangle where A lies on x-axis, C lies on y-axis and B lies on the curve x^2/3 + y^2/3 = a^2/3 , O is the origin.
Find the coordinates of B so that the area of OABC is maximum.

Answer 1

Hi Prashant, 

 

I am making some assumptions here: 1) you make no mention of calculus so I assume you do not have that at your disposal to determine when a function has a maximum. 2) Perhaps you know that for a rectangle satisfying the given conditions to generate a maximum area it must be a square? 

 

Assuming #2, we can solve the original equation for y as a function of x. y = [a^2/3 - x^2/3]^3/2, where a is a constant. See this graph to get the idea (a y value for a given x represents the height of the rectangle, x represents the width):

 

https://www.desmos.com/calculator/hvkgd9f1kv  (try the sliders for a and b).

 

For the rectangle to be a square, then x = y. 

 

Solving this for x is the key then:

 

x =  [a^2/3 - x^2/3]^3/2

 

Consider raising both sides to the 2/3 power. Transpose terms containing x to left side (2x^2/3 = a^2/3).

 

Divide both sides by 2, then raise both sides to 3/2 power. Should get something like x = a√2/4

 

So the constant a can be used to determine the x/y values that will generate the rectangle of maximum area.