Rectangle inscribed in an ellipse.

[tysonk]

Find the area of the largest rectangle that can be inscribed (with sides parallel to the axes in the ellipse).
x^2/a^2 +y^2/b^2 = 1

I came across the above problem and am not sure how to proceed with it. I drew the ellipse with the inscribed rectangle and tried repositioning the ellipse so that the corner of the rectangle is placed at the origin.

Then the corners have coordinates.
(0,0) , (p, 0), (p, q), (0, q)
A = pq so we want the maximum A however I'm not sure where to go from here.

 

[noleguy33]

Solve for y, plug that value into your area equation... then take the derivative and set it to zero. Solve for x and then plug that value for x back into the area equation(the one in which you've substituted for the y value) and then simplify.

Hopefully someone can back me up, but I believe that's the best way to do it.

 

[tysonk]

When finding the derivative of that then I'll have a dy/dx term? Also would I have to times that area by 2 since the ellipse equation is such that the center of the ellipse is at the origin...

 

[HallsofIvy]

When finding the derivative of that then I'll have a dy/dx term? Also would I have to times that area by 2 since the ellipse equation is such that the center of the ellipse is at the origin...

If you are referring to your "(0,0) , (p, 0), (p, q), (0, q)" then you would multiply by 4 to get the area of the entire rectangle since that is only in the first quadrant.