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# Ellipse Tangent

The normal to an ellipse at a point intersects the ellipse at another point . The angle corresponding to can be found by solving the equation

 (1)

for , where and . This gives solutions

 (2)

where

 (3)

of which gives the valid solution. Plugging this in to obtain then gives

 (4) (5) (6)

To find the maximum distance, take the derivative and set equal to zero,

 (7)

which simplifies to

 (8)

Substituting for and solving gives

 (9) (10)

Plugging these into then gives

 (11)

This problem was given as a Sangaku problem on a tablet from Miyagi Prefecture in 1912 (Rothman 1998). There is probably a clever solution to this problem which does not require calculus, but it is unknown if calculus was used in the solution by the original authors (Rothman 1998).

Ellipse

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## References

Rothman, T. "Japanese Temple Geometry." Sci. Amer. 278, 85-91, May 1998.

Ellipse Tangent

## Cite this as:

Weisstein, Eric W. "Ellipse Tangent." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EllipseTangent.html