Finding the Equation of Ellipse
We now examine the method of finding the equation of an ellipse given certain identifying information.
Central Ellipses
The standard form for a central ellipse depends on the orientation of the ellipse. The equations and attributes can be summarized in the table below, given the following:
- The parameter $a$a is the length of the semi-major axis.
- The parameter $b$b is the length of the semi-minor axis.
- The parameter $c$c is the distance from the center to each focus.
| Orientation | Horizontal Major Axis | Vertical Major Axis |
|---|---|---|
| Standard form | $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$x2a2+y2b2=1 | $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$x2b2+y2a2=1 |
| Center | $\left(0,0\right)$(0,0) | $\left(0,0\right)$(0,0) |
| Foci | $\left(c,0\right)$(c,0) and $\left(-c,0\right)$(−c,0) | $\left(0,c\right)$(0,c) and $\left(0,-c\right)$(0,−c) |
| Vertices | $\left(a,0\right)$(a,0) and $\left(-a,0\right)$(−a,0) | $\left(0,a\right)$(0,a) and $\left(0,-a\right)$(0,−a) |
| Covertices | $\left(0,b\right)$(0,b) and $\left(0,-b\right)$(0,−b) | $\left(b,0\right)$(b,0) and $\left(-b,0\right)$(−b,0) |
| Major axis | $y=0$y=0 | $x=0$x=0 |
| Minor axis | $x=0$x=0 | $y=0$y=0 |
Notice that by this definition, it is always true that $a>b$a>b. It is also true that the parameters $a$a, $b$b, and $c$c have the relationship $c^2=a^2-b^2$c2=a2−b2.
Translated Ellipses
If an ellipse is translated horizontally or vertically from the center, the parameter $a$a, $b$b, and $c$c still have the same meaning. However, we must take into account that the center of the ellipse has moved. Given the following definitions for $h$h and $k$k,
- The parameter $h$h denotes the translation in the horizontal direction from $0,0$0,0
- The parameter $k$k denomes the translation in the vertical direction from $0,0$0,0
The table below summarizes the standard form of an ellipse in both orientations.
| Orientation | Horizontal Major Axis | Vertical Major Axis |
|---|---|---|
| Standard form | $\frac{\left(x-h\right)^2}{a^2}+\frac{\left(y-k\right)^2}{b^2}=1$(x−h)2a2+(y−k)2b2=1 | $\frac{\left(x-h\right)^2}{b^2}+\frac{\left(y-k\right)^2}{a^2}=1$(x−h)2b2+(y−k)2a2=1 |
| Center | $\left(h,k\right)$(h,k) | $\left(h,k\right)$(h,k) |
| Foci | $\left(h+c,k\right)$(h+c,k) and $\left(h-c,k\right)$(h−c,k) | $\left(h,k+c\right)$(h,k+c) and $\left(h,k-c\right)$(h,k−c) |
| Vertices | $\left(h+a,k\right)$(h+a,k) and $\left(k-a,k\right)$(k−a,k) | $\left(h,k+a\right)$(h,k+a) and $\left(0,-a\right)$(0,−a) |
| Covertices | $\left(h,k+b\right)$(h,k+b) and $\left(h,k-b\right)$(h,k−b) | $\left(h+b,k\right)$(h+b,k) and $\left(h-b,k\right)$(h−b,k) |
| Major axis | $y=k$y=k | $x=h$x=h |
| Minor axis | $x=h$x=h | $y=k$y=k |
Essentially, the information is the same as the central ellipse. But the values of $h$h and $k$k are added to the $x$x and $y$y values (respectively) for each characteristic.
Worked Examples
question 1
Find the equation, in standard form, of the ellipse with $x$x-intercepts $\left(\pm3,0\right)$(±3,0) and $y$y-intercepts $\left(0,\pm7\right)$(0,±7).
question 2
Find the equation, in standard form, of the ellipse with foci $\left(\pm6,0\right)$(±6,0) and a minor axis of length $6$6.
question 3
Find the equation, in standard form, of the ellipse with center $\left(1,5\right)$(1,5), a horizontal major axis of length $16$16 and a minor axis of length $10$10.