Identify Characteristics of Ellipses

An ellipse centred at the origin has either one of two equations. An ellipse may have its major axis (longest diameter) along the $x$x-axis and its minor axis (shortest diameter) along the $y$y-axis. In this case, the equation of the ellipse is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$x2a2​+y2b2​=1 where:

• $a$a is half the length of the major axis
• $b$b is half the length of the minor axis

Half of the major axis is referred to as the semi-major axis, and half of the minor axis is referred to as the semi-minor axis. So we can say that $a$a is the length of the semi-major axis, and $b$b is the length of the semi-minor axis.

Graph of ellipse with equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$x2a2​+y2b2​=1 where $a>b$a>b.

 

If we swap the position of $a$a and $b$b, we get the equation $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$x2b2​+y2a2​=1 where $a>b$a>b with the following graph. We call these two equations the standard form of the ellipse centred at the origin.

Graph of ellipse with equation $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$x2b2​+y2a2​=1 where $a>b$a>b.

 

We call the endpoints of the major axis the vertices, and the endpoints of the minor axis the co-vertices. In the special case that the ellipse is centred at the origin, then these points represent the $x$x- and $y$y-intercepts.

Vertices and co-vertices of an ellipse.

 

If the vertices lie on the $x$x-axis, then their coordinates are $\left(-a,0\right)$(−a,0) and $\left(a,0\right)$(a,0). If they lie on the $y$y-axis then their coordinates are $\left(0,-a\right)$(0,−a) and $\left(0,a\right)$(0,a).

Similarly, if the co-vertices lie on the $x$x-axis, then they have coordinates $\left(-b,0\right)$(−b,0) and $\left(b,0\right)$(b,0). If they lie on the $y$y-axis then their coordinates are $\left(0,-b\right)$(0,−b) and $\left(0,b\right)$(0,b).

We might also be curious about identifying the foci of a given ellipse. From our definition of the ellipse, the foci represent two points on the major axis such that their sum of the distances to the ellipse is always constant.

Construction of the ellipse from two foci.

 

Recall from the definition of the ellipse, we start with the foci $\left(-c,0\right)$(−c,0) and $\left(c,0\right)$(c,0) and arrive at the equation of the ellipse where $c^2=a^2-b^2$c2=a2−b2.

Similarly, if we started with the foci $\left(0,-c\right)$(0,−c) and $\left(0,c\right)$(0,c), then we can obtain the same equation $c^2=a^2-b^2$c2=a2−b2.

Alternatively, knowing the values of $a$a and $b$b, we can find the coordinates of the two foci using the equation $c^2=a^2-b^2$c2=a2−b2.

Worked example

Consider the ellipse with the equation $\frac{x^2}{9^2}+\frac{y^2}{4^2}=1$x292​+y242​=1.

What are the coordinates of the two vertices?

Think: The vertices are the endpoints of the major axis. The major axis lies on the $x$x-axis, so the vertices will be of the form $\left(-a,0\right)$(−a,0) and $\left(a,0\right)$(a,0).

Do: Since $a=9$a=9, the vertices are $\left(-9,0\right)$(−9,0) and $\left(9,0\right)$(9,0).

 

What are the coordinates of the two co-vertices?

Think: The co-vertices are the endpoints of the minor axis. The minor axis lies on the $y$y-axis, so the vertices will be of the form $\left(0,-b\right)$(0,−b) and $\left(0,b\right)$(0,b).

Do: Since $b=4$b=4, the co-vertices are $\left(0,-4\right)$(0,−4) and $\left(0,4\right)$(0,4).

 

What are the coordinates of the foci?

Think: The distance $c$c from the centre to a focus of the ellipse is related to the semi-major and semi-minor axes by the equation $c^2=a^2-b^2$c2=a2−b2. We can substitute the known values for $a$a and $b$b, and then solve for $c$c.

Do:

$c^2$c2	$=$=	$a^2-b^2$a2−b2	(Writing the equation)
$c^2$c2	$=$=	$9^2-4^2$92−42	(Substitution)
$c^2$c2	$=$=	$81-16$81−16	(Simplifying the squares)
$c^2$c2	$=$=	$65$65	(Simplifying the subtraction)
$c$c	$=$=	$\sqrt{65}$√65	(Taking the square root)

The ellipse is centred at the origin, so the coordinates of the foci are $\left(-\sqrt{65},0\right)$(−√65,0) and $\left(\sqrt{65},0\right)$(√65,0).

Reflect: How might the coordinates of the foci change if the equation of the ellipse was $\frac{x^2}{4^2}+\frac{y^2}{9^2}=1$x242​+y292​=1?

Practice questions

question 1

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