The Focal Property of an Ellipse
An ellipse is a closed curve that can be described as the locus of points for which the sum of the distances to two given points (called foci) is a constant.
An ellipse has two axes of symmetry.
The major axis is the line through (or line segment between) the two points most distant from the center (vertices).
The minor axis is the line through (or line segment between) the two points least distant from the center (co-vertices).
The line segments from the center to each vertex are called the semi-major axes, and the line segments from the center to each co-vertex are called the semi-minor axes.
The foci of an ellipse, E and F, lie on the ellipse's major axis and are equidistant from the center. The sum of the distances from any point P on the ellipse to these two foci is equal to the length of the major axis.
The general equation for a horizontal ellipse (assuming a>b ) is:
x−h2a2 + y−k2b2 = 1,
while the general equation for a vertical ellipse is:
x−h2b2 + y−k2a2 = 1,
where h, k is the center, a is the length of the semi-major axis, and b is the length of the semi-minor axis.
Derivation of the General Equation from the Focal Property
For simplicity, let's say that the ellipse is horizontal, centered at 0, 0 with the following foci: E at −c, 0 and F at c, 0. Thus, the distance from each focus to the center is c.
The distance from a general point Px, y on the ellipse to E is given by PE = x − −c2+y − 02 = x+c2+y2.
The distance from P to F is given by PF = x − c2+y −02= x − c2+y2.
Looking at the case in which P is a vertex of the ellipse and adding up the distances from this vertex to each focus, we see that the sum of these distances is 2⁢a.
Thus, we know that:
x+c2+y2 = 2⁢a − x−c2+y2
x+c2+y2 = 2 a − x−c2+y22
x2+2⁢c⁢x+c2+y2 = 4⁢a2−4⁢a⁢x−c2+y2 + x2 − 2 c x + c2 + y2
2⁢c x = 4 a2−4⁢a⁢x−c2+y2 − 2 c x
4⁢c x − 4 a2 = −4 a x−c2+y2
c x − a2 = −a x−c2+y2
c x − a22 = a2⁢x−c2+y2
c2⁢x2−2⁢a2⁢c⁢x+ a4 = a2⁢x2−2 a2⁢c⁢x+ a2⁢c2 +a2⁢y2
c2⁢x2+ a4 = a2⁢x2+ a2⁢c2 +a2⁢y2
a4−a2⁢c2=a2⁢x2−c2⁢x2 + a2⁢y2
Considering the case when P is the co-vertex located at 0,b, we see that b2 + c2 = a2, so b2 = a2 − c2. Substituting b2, we get:
a2⁢b2 = b2⁢x2+ a2⁢y2
a2⁢b2a2⁢b2 = b2⁢x2+ a2⁢y2a2⁢b2
1 = x2a2 + y2b2
This is the standard equation for an ellipse centered at 0, 0 with horizontal semi-major axis a and semi-minor axis b.
Click to add two points to the plot below to set the foci, E and F. Then, choose the "Plot Points" radio button and click on the graph to plot the points creating an ellipse. Select the "Show Ellipse" check box to see the actual curve and its equation. Click "Reset" to reset the plot.
Distance from Focus E to Latest Point =
Distance from Focus F to Latest Point =
Sum of Distances =
Equation of Ellipse:
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