Tangents to a Rotating Ellipse

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This Demonstration plots the tangents and the locus of the points of tangency to a rotating ellipse starting from a point on the axis that you can drag along the axis.
Contributed by: Erik Mahieu (November 2012)
Open content licensed under CC BY-NC-SA
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Details
The equation of an ellipse with semimajor axis and eccentricity
rotated by
radians about its center at the origin is
.
The equation of a line through the point and cutting the
axis at an angle
is
.
Solving these two equations simultaneously gives the two points of intersection of the line with the rotating ellipse. The line is tangent if these two points coincide. This happens with the angle that satisfies the equation that sets the two
coordinates of the intersection points equal,
.
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