Tangent to an ellipse - Math Open Reference

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Tangent to an ellipse

A line on the outside of an ellipse that touches it at just one point

Try this Drag any orange dot. Note the tangent line touches at just one point.

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The blue line on the outside of the ellipse in the figure above is called the "tangent to the ellipse". Another way of saying it is that it is "tangential" to the ellipse. (Pronounced "tan-gen-shull"). It is a similar idea to the tangent to a circle.

The line barely touches the ellipse at a single point. If the line were closer to the center of the ellipse, it would cut the ellipse in two places and would then be called a secant. In fact, you can think of the tangent as the limit case of a secant. As the secant line moves away from the center of the ellipse, the two points where it cuts the ellipse eventually merge into one and the line is then the tangent.

Properties

The tangent line always makes equal angles with the generator lines.
Recall from the definition of an ellipse that there are two 'generator' lines from each focus to any point on the ellipse, the sum of whose lengths is a constant.
Try this: In the figure above click reset then drag any orange dot. No matter how you reshape the ellipse, the tangent line makes equal angles to each generator line a,b.
The perpendicular from the tangent through the contact point bisects the angle between generator lines.
Try this: In the figure above click 'show perpendicular'. The perpendicular line you now see bisects the angle between the two generator lines a and b. Move the contact point or either focus point and convince yourself this is always true.
A circle is just a special case of an ellipse
Try this: In the figure above click 'show perpendicular'. Then drag either focus on top of the other. The ellipse is now a circle. As you drag the contact point you will see that the tangent behaves in exactly the same way as seen in tangent to a circle.

Other ellipse topics