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What is the correct path formula for high school mathematics and plane analytic geometry problems?
The path of an ellipse is the length of the line segment obtained by the intersection of a straight line perpendicular to the long axis and the ellipse.

So use c instead of x in the elliptic equation,

Y 1 = b 2/a and y2 =-b/a can be obtained.

So the length of the path is y1-y2 = 2b 2/a.

Where b 2 stands for the square of B.