Rectangle inscribed in an ellipse.

In summary, to find the area of the largest rectangle that can be inscribed in an ellipse with sides parallel to the axes, we can reposition the ellipse so that the corner of the rectangle is placed at the origin. Then, using the coordinates of the corners, we can set up an area equation and take the derivative to find the maximum area. This will involve solving for one variable and then substituting it back into the equation. The final area will need to be multiplied by 4 to account for all four quadrants of the ellipse.
  • #1
tysonk
33
0
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.
 
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  • #2
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.
 
  • #3
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...
 
  • #4
tysonk said:
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.
 

1. What is a rectangle inscribed in an ellipse?

A rectangle inscribed in an ellipse is a rectangle that is drawn inside an ellipse in such a way that all four of its vertices touch the ellipse. This means that the sides of the rectangle are tangent to the ellipse, and the rectangle is completely contained within the ellipse.

2. How is the area of a rectangle inscribed in an ellipse calculated?

The area of a rectangle inscribed in an ellipse can be calculated using the formula A = ab, where a and b are the lengths of the rectangle's semi-major and semi-minor axes. These values can be determined by using the formula for the semi-major axis (a = length of ellipse's major axis / 2) and the formula for the semi-minor axis (b = length of ellipse's minor axis / 2).

3. What is the relationship between the lengths of the sides of a rectangle inscribed in an ellipse?

The sides of a rectangle inscribed in an ellipse have a special relationship known as the Pythagorean theorem. This means that the square of the length of one side of the rectangle is equal to the sum of the squares of the other two sides. In other words, a² + b² = c², where a and b are the lengths of the rectangle's sides and c is the length of the rectangle's diagonal.

4. Can a rectangle be inscribed in any ellipse?

Yes, a rectangle can be inscribed in any ellipse, as long as the rectangle's dimensions are smaller than those of the ellipse. This means that the rectangle's length and width must be less than the length and width of the ellipse's major and minor axes.

5. What are the properties of a rectangle inscribed in an ellipse?

A rectangle inscribed in an ellipse has several interesting properties, including that its diagonals are equal in length, its opposite sides are parallel, and its opposite angles are equal. Additionally, the area of the rectangle is always less than or equal to half the area of the ellipse.

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