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The answer will depend on what, if anything, the line segments have to do with the ellipse.
No it cannot. A polygon is a plane space enclosed by straight lines. An ellipse consists of a curved line, not straight lines.
yes half of that line is called the radius (used to calculate the area of a circle)
The shortest path is a line perpendicular to the given line that passes through the given point.
If the eccentricity was 0 the ellipse would instead be a circle, and if the eccentricity was 1 it would be a straight line segment.