Find the area of the greatest square that can be inscribed in the ellipse x²/a² + y²/b² = 1.
Please help!
Please help!
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I don't know what they mean by "greatest" square. Seems to me that only one square can be inscribed in an ellipse.
The ellipse x²/a² + y²/b² = 1 is centered at the origin.
The corners of the inscribed square intersect the ellipse when x = y:
x²/a² +x²/b² = 1
b²x²/(a²b²) + a²x²/(a²b²) = 1
(b² + a²)x²/(a²b²) = 1
x² = a²b²/(a² + b²)
Each side of the square has length 2x, so the area of the square is 4x² = 4a²b²/(a² + b²).
The ellipse x²/a² + y²/b² = 1 is centered at the origin.
The corners of the inscribed square intersect the ellipse when x = y:
x²/a² +x²/b² = 1
b²x²/(a²b²) + a²x²/(a²b²) = 1
(b² + a²)x²/(a²b²) = 1
x² = a²b²/(a² + b²)
Each side of the square has length 2x, so the area of the square is 4x² = 4a²b²/(a² + b²).