Applications of Ellipses

Below are two images with the important features of an ellipse:

Horizontally aligned ellipse	Vertically aligned ellipse

 

The standard form for a central ellipse depends on the orientation of the ellipse.  The equations and attributes are summarized in the table below, given the following:

• $a$a is the length of the semi-major axis.
• $b$b is the length of the semi-minor axis.
• $c$c is the distance from the center to each focus.

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.

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)
Co-vertices	$\left(0,b\right)$(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

 

Translated ellipses

If an ellipse is translated horizontally or vertically from the origin, the parameter $a$a, $b$b, and $c$c still have the same meaning. However, we must take into account that the centre of the ellipse has moved.  Given the following definitions for $h$h and $k$k,

• $h$h denotes the translation in the horizontal direction from $\left(0,0\right)$(0,0).
• $k$k denotes the translation in the vertical direction from $\left(0,0\right)$(0,0).

An ellipse translated horizontally by $h$h and vertically by $k$k.

The table below summarises 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 $\left(h-a,k\right)$(h−a,k)	$\left(h,k+a\right)$(h,k+a) and $\left(h,k-a\right)$(h,k−a)
Co-vertices	$\left(h,k+b\right)$(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.

 

Application to orbits

Orbits around a planetary body are described by the conic sections (circle, ellipse, parabola and hyperbola).

Elliptical orbits around the Sun. Not to scale.

The orbits of the planets around the Sun and the moon around the Earth are all elliptical (some are very close to circular though). The larger mass being orbited will be at one of the foci of the ellipse. For example, the Sun will be at one of the foci of the Earth's ellipse.

 

Worked example

At the closest point in its orbit (Perihelion) Mercury is $0.31$0.31 AU away from the Sun. At its furthest point in its orbit (Aphelion) it is $0.36$0.36 AU from the Sun. Find the equation of Mercury's orbit around the Sun. Give your values to two decimal places in AU.

Think: The equation of the ellipse will be in the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$x2a2​+y2b2​=1. We can find the values of $a$a and $c$c using the geometry of the ellipse. The value of $b$b can be found given that $c^2=a^2-b^2$c2=a2−b2.

From the image above we can see that the value of $a$a will be $\frac{0.31+0.36}{2}$0.31+0.362​ and $c$c will be $\frac{0.36-0.31}{2}$0.36−0.312​.

Do: Solve for $a$a and $c$c and then sub into equation for $b$b.

$a$a	$=$=	$\frac{0.31+0.36}{2}$0.31+0.362​
	$=$=	$0.335$0.335
$c$c	$=$=	$\frac{0.36-0.31}{2}$0.36−0.312​
	$=$=	$0.025$0.025
$b$b	$=$=	$\sqrt{a^2-c^2}$√a2−c2
	$=$=	$\sqrt{0.335^2-0.025^2}$√0.3352−0.0252
	$\approx$≈	$0.334$0.334 (3 d.p.)

If we create our $xy$xy plane so the major axis is aligned with $x$x-axis and the $y$y-axis is aligned with the minor axis we can determine the equation of the orbit.

$\frac{x^2}{0.335^2}+\frac{y^2}{0.334^2}=1$x20.3352​+y20.3342​=1

 

 

Practice Questions

QUestion 1

Question 2

Question 3