Question

The perimeter of a rectangle having the largest area that can be inscribed in the ellipse

Answer 1

Answer:

Equation of ellipse, (x2/a2) + (y2/b2) = 1

Thinking of the area as a function of x, we have

dA/dx = 4xdy/dx + 4y

Differentiating equation of ellipse with respect to x, we have

2x/a2 + (2y/b2)dy/dx = 0,

so,

dy/dx = -b2x/a2y,

and

dAdx = 4y – (4b2x2/a2y)

Setting this to 0 and simplifying, we have y2 = b2x2/a2.

From equation of ellipse we know that,

y2=b2 – b2x2/a2

Thus, y2=b2 – y2, 2y2=b2, and y2b2 = 1/2.

Clearly, then, x2a2 = 1/2 as well, and the area is maximized when

x= a/√2 and y=b/√2

So the maximum area Area, Amax = 2ab