Ellipse is the plane curve that contains all the points that the sum of distance from two focal points all the same. That is, for two focal points \(F_{1}, F_{2}\), \(F_{1}P + F_{2}P = d > 0\). Usually we denoted the sum of distance \(d=2a\), and the distance between two focal, called focal distance or linear eccentricity, as \(2c\).
The equation of standard ellipse, which centered at the origin(that is, the two focal points are put at \(x\) axis symmetrically in \((c, 0), (-c, 0)\) respectively), computed as
\[ \sqrt{(x+c)^2 + y^2} + \sqrt{(x-c)^2 + y^2} = 2a \]
That is,
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
where we define \(b^2 = a^2 - c^2\).
Obviously, this standard ellipse intersects with \(x, y\) axis as \((\pm a, 0), (0, \pm b)\) respectively, hence \(a, b\) are named as the semi-major and semi-minor axis.
The length of chord that generates from standard form equation of ellipse and line can be expressed as:
\[ d = \frac{2\sqrt{a^2b^2(A^2+B^2)(a^2A^2+b^2B^2+C^2)}}{a^2A^2+b^2B^2} \]