Height of Line Segment on an Ellipsis I'm trying to find the equation for getting the height of the black line I show in the image below. The end point of the black line is the intersection point between the width of the square below and the ellipse itself. I know the value of the width and the height of the ellipse, and the width of the square. I need to know the value of the height of the line I show below.
Also, the square is always centered below the ellipse.
My ellipse can be either vertical or horizontal, how would that affect the result?
Thank you very much.


Let's suppose that the width of the ellipse is $2w$ and the width of the rectangle is $2u$. Let's also suppose that the height of the ellipse is $2h$. Then if we scale the horizontal axis by a factor of $h/w$, the ellipse becomes a circle of radius $h$ and the rectangle half-width becomes $uh/w$ instead of $u$. 
So you're asking for the distance from the tangent to the bottom of that circle -- that's at $y = -h$ -- to the point above the right-hand side of the rectangle, which is at $x = uh/w$. Well, the points on the circle (assuming we center it at the origin) satisfy
$$
x^2 + y^2 = h^2
$$
so we have
\begin{align}
(uh/w)^2 + y^2 &= h^2 \\
\frac{u^2h^2}{w^2} + y^2 &= h^2 \\
y^2 &= h^2 - \frac{u^2h^2}{w^2} \\
y^2 &= h^2( 1  - \frac{u^2}{w^2} )\\
y &= -h \sqrt{ 1  - \frac{u^2}{w^2}},
\end{align}
where in the last step I've chosen the negative square root because we want the lower value of $y$. 
The difference between this and the $y$-value at the bottom, namely $y = -h$, is
\begin{align}
-h \sqrt{ 1  - \frac{u^2}{w^2}} - (-h) &= h\left(1 - \sqrt{1 - \frac{u^2}{w^2}}\right)
\end{align}
When $u = 0$, i.e., when the rectangle is just a very thin vertical line, this gives
\begin{align}
 h\left(1 - \sqrt{1 - \frac{u^2}{w^2}}\right) &= h\left(1 - \sqrt{1 - \frac{0^2}{w^2}}\right)\\
&= h\left(1 - \sqrt{1}\right)\\
&= h\left(0\right)\\
&= 0,
\end{align}
which looks right, and when $u = w$ (so that the square is the same width as the ellipse), it gives
\begin{align}
 h\left(1 - \sqrt{1 - \frac{u^2}{w^2}}\right) &= h\left(1 - \sqrt{1 - \frac{w^2}{w^2}}\right)\\
&= h\left(1 - \sqrt{0}\right)\\
&= h\left(1\right)\\
&= h,
\end{align}
which also looks right. So I'm fairly confident that I didn't mess up the algebra. :)
