An ellipse is the set of points in a plane where the sum of the distances from any point on the curve to two fixed points, called the foci 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 midpoint of the major axis, serving as a focal point for its geometry.

Additionally, every ellipse has two axes of symmetry. One runs through both foci, known as the transverse axis, while the other, perpendicular to the first, is called the conjugate axis. Depending on their orientation, the corresponding ellipse can be 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, while $(h,k)$ denotes the center of the ellipse.