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Point of Intersection of an Ellipse with a Line

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This Demonstration shows a ruler and compass construction of an intersection of an ellipse with semiaxes and with a line.

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Suppose the ellipse (in blue) has the equation

and that the line (in red) has the equation

.

The linear transformation transforms the ellipse into the circle and the line into a new line with the equation . We can construct this line with slope and its intersection with the circle. Let be one of the intersection points. Then is one of the intersections of the ellipse with the red line. To construct the point , we must find the intersection point of the line and the line . The point is the intersection of the line and the perpendicular line at on .

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Contributed by: Izidor Hafner (March 2018)
Open content licensed under CC BY-NC-SA

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