S=π(pi)×a×b(where a,b are the lengths of the half-length axis and half-short axis of the ellipse respectively).

Or S=π(pi)×A×B/4(where A,B are the lengths of the long and short axes of the ellipse respectively).

Extension

Area of cross-shading of circle and ellipse

The ratio of the area of the circle to the area of the ellipse is cosθ, then cosθ=πR^2/S=2R/2a, and the short axis b of the ellipse is the radius R of the base of the cylinder,i.e., R=b, so S=πR^2*a/R=πaR=πab

Equation for slope of ellipse

Slope of tangent line to a point (x, y) on ellipse x^2/a^2+y^2/b^2=1 is -(b^2)X/(a^2)y

Area of triangle formed by a point (x, y) on ellipse and two foci S=b^2*tan(α/2) α is the angle of the line between a point (x, y) and two foci

Area of triangle formed by point (x, y) and two foci S=b^2*tan(α/2) α is the angle of the line connecting point (x, y) and two foci

Reference:

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