An ellipse is a set of points in a plane where the sum of the distances from any point on the curve to two fixed points, each called the focus of the ellipse, is constant. In the provided figure, the sum of the distances from the foci $F_1$ and $F_2$ to a point $P$ on the ellipse is $8.$ The vertices of an ellipse are connected by the major axis, while the co-vertices are linked by the minor axis. The center of the ellipse lies at the intersection of the axes, serving as a focal point for its geometry. Every ellipse has two axes of symmetry. One runs through both foci, known as the transverse axis. The other runs perpendicular to the first and is called the conjugate axis. Depending on their orientation, the corresponding ellipse is called horizontal or vertical.

Property	Horizontal Ellipse	Vertical Ellipse
General Equation	$\frac{(x-h{)}^2}{a^2}+\frac{(y-k{)}^2}{b^2}=1$	$\frac{(x-h{)}^2}{b^2}+\frac{(y-k{)}^2}{a^2}=1$
Transverse Axis	Horizontal	Vertical
Vertices	$(h\pm a,k)$	$(h,k\pm a)$
Foci	$(h\pm c,k)$	$(h,k\pm c)$
$c\text{-value}$	$c^2=a^2-b^2$	$c^2=a^2-b^2$

In these equations, $a$ and $b$ represent half the lengths of the major and minor axes, respectively. The center of the ellipse is denoted by $(h,k).$ The value $c$ is related to the eccentricity of the ellipse.