Question

find the area of the largest rectangle that can be inscribed in the ellipse x 2 a 2 y 2 b 2 = 1 x2a2 y2b2=1 .

Answer 1

The area of the largest rectangle inscribed in an ellipse is 4ab, where a is the semi-major axis and b is the semi-minor axis of the ellipse.

Step-by-step explanation:

To find the area of the largest rectangle that can be inscribed in the ellipse, we need to consider the semi-major axis (denoted by a) and the semi-minor axis (denoted by b) of the ellipse.

The area of the rectangle will be greatest when it is aligned with the axes of the ellipse.

So, the length of the rectangle will be 2a and the width of the rectangle will be 2b.

Therefore, the area of the largest rectangle inscribed in the ellipse is A = 2a * 2b = 4ab.

Answer 2

The area of the largest rectangle that can be inscribed in an ellipse with equation x^2/a^2 + y^2/b^2 = 1 is 2ab.

Step-by-step explanation:

To find the area of the largest rectangle that can be inscribed in the ellipse x2/a2 + y2/b2 = 1, we first need to understand the properties of the ellipse. The equation of the ellipse represents that the sum of the squared distances from any point on the ellipse to the foci is always constant. It can be shown that the largest rectangle that can be inscribed in the ellipse is one where the diagonal of the rectangle is the major axis of the ellipse.

For the given ellipse with equation x2/a2 + y2/b2 = 1, the length of the major axis is 2a and the length of the minor axis is 2b. Therefore, the area of the largest inscribed rectangle is 2ab.