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).