I'm looking for an equation that can help me determine the length of the minor axis.
I know the length of the major axis and have the Cartesian coordinates of a point somewhere on the ellipse.
How can I use these to get the length of the minor axis?
[Math] Equation for getting the length of the minor axis (of an ellipse)
analytic geometryconic sectionsgeometry
Best Answer
Added: In a comment OP states that "The major axis is on the y-axis and the minor axis is on the x-axis."
The equation of an ellipse whose major and minor axis are respectively on the $y$ and $x$-axis is
$$\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1,\qquad (\ast )$$
where $a$ is the semimajor axe and $b$ is the semiminor axe. You are given $% 2a$ and you need to find $2b$. Let the coordinates of the given point be $% (x_{1},y_{1})$. Since it is on the ellipse, its coordinates must satisfy $% (\ast )$
$$\frac{x_{1}^{2}}{b^{2}}+\frac{y_{1}^{2}}{a^{2}}=1.\qquad (\ast \ast )$$
Clearing denominators and then dividing by $y_{1}^{2}-a^{2}$ we get
$$a^{2}x_{1}^{2}+b^{2}y_{1}^{2}=a^{2}b^{2}\Leftrightarrow \left( y_{1}^{2}-a^{2}\right) b^{2}=-a^{2}x_{1}^{2}\Leftrightarrow b^{2}=-\frac{% a^{2}x_{1}^{2}}{y_{1}^{2}-a^{2}}=\frac{a^{2}x_{1}^{2}}{a^{2}-y_{1}^{2}}.$$
Since $a^{2}-y_{1}^{2}\geq 0$ and $b>0$, we obtain
$$b=\frac{a|x_{1}|}{\sqrt{a^{2}-y_{1}^{2}}}.\qquad (\ast \ast \ast )$$
The length of the minor axe is $2b$.