Final answer:
The area of the largest rectangle that can be inscribed in an ellipse with the equation x²/a² + y²/b²= 1 is represented by the formula 4ab. This maximum area is attained when the rectangle's vertices touch the ellipse at its widest points along the major and minor axes.
Step-by-step explanation:
To find the area of the largest rectangle that can be inscribed in an ellipse with the equation x²/a² + y²/b² = 1, we need to use the properties of an ellipse and the characteristics of inscribed rectangles. Given the semi-major axis a and the semi-minor axis b of the ellipse, any rectangle inscribed within this ellipse will have vertices at (+/- x, +/- y), which satisfy the ellipse equation. For such a rectangle, the lengths of the sides will be 2x and 2y.
To find the maximum area, we set up the problem using the ellipse equation and implicit differentiation, or we recognize that the area of the rectangle will be maximized when its vertices lie on the ellipse. The maximum occurs when x = a and y = b, as this is when the rectangle's area formula A = 2x * 2y yields the largest value. Therefore, the area of the largest inscribed rectangle is 4ab, which corresponds to option (a) 4ab.