Ellipse Graph

As we know, we can write an equation of an ellipse given its graph, we can also graph an ellipse given its equation. The equations help us to draw an ellipse on a graph with the x & y coordinate planes. This article will demonstrate how we can graph an ellipse from a given equation both in standard and general form.

To graph an ellipse, we first need to find out its center, foci, vertices, and co-vertices. The equations help us to find these parameters. From our ellipse article, we know that ellipses have 2 orientations— horizontal or vertical. The equations in both forms also help us identify its orientation.

How to Graph an Ellipse in Standard Form: Not Centered at Origin

Here we will see how an ellipse equation in standard form, centered at (h,k) is plotted on a graph.

For Horizontal Ellipse

The standard form of an equation for a horizontal ellipse (foci on major axis) centered at (h,k) is:

 (x – h)²/a² + (y – k)²/b² = 1, (a>b)

Now, let us learn to plot an ellipse on a graph using an equation as in the above form.

Let’s take the equation (x – 1)²/9 + (y – 2)²/4= 1 and identify whether it is a horizontal or vertical ellipse. We will also label the center, vertices, co-vertices, and foci.

From the given equation, it is clear that the ellipse is horizontal, seeing the positions of a² and b² as in the denominators. So, the foci & vertices must be on the x-axis.

Comparing the given equation with standard form (x-h)²/a² + (y-k)²/b² = 1, we get:

Center: (h, k); h =1, k = 2, so Center: (1, 2)

Vertices: (h ± a, k) = (4,2) & (-2,2)

Co-vertices: (h, k ± b) = (1,4) & (1,0)

Length of Major Axis: 2a, Length of Minor Axis: 2b

∴ a = 3, b = 2

Foci where F² = a² – b² , F = (h±F, k) {F = distance from center to the foci or focal length}

Equation of focal length F is F² = a² – b²  = 9-4 = 5, F = √5 ≈ 2.236

∴ Foci = (1±√5, 2) = (3.236,2) & (-1.236,2)

Now let’s graph the ellipse.

Now let’s solve some more examples when the ellipse equation is in standard form, not centered at the origin.

For Vertical Ellipse

The standard form of an ellipse is for a vertical ellipse (foci on minor axis) centered at (h,k)

(x – h)²/b² + (y – k)²/a² = 1 (a>b)

Now, let us learn to plot an ellipse on a graph using an equation as in the above form.

Let’s take the equation x²/25 + (y – 2)²/36 = 1 and identify whether it is a horizontal or vertical ellipse. We will also label the center, vertices, co-vertices, and foci.

From the given equation, it is clear that the ellipse is vertical, seeing the positions of a² and b² as in the denominators. So, the foci & vertices must be on the y-axis.

Comparing the given equation with standard form (x-h)²/b² + (y-k)²/a² = 1, we get:

∴ h =0, k = 2, Center (h, k)  = (0,2)

Length of Major Axis: 2a, Length of Minor Axis: 2b

a = 6, b = 5

Vertices: (h, k ± a) = (0,8) & (0,-4)

Co-vertices: (h ± b, k) = (5,2) & (-5,2)

Equation of focal length F is F² = a² – b²  = 36-25 = 11, F = √11 ≈ 3.316 ≈ 3.32

∴ Foci = (h, k±F) = (0, 2±√11) = (0, 2±3.32) = (0,5.32) & (0,-1.32)

Now let’s graph the ellipse.

How to Graph an Ellipse in General Form: Not Centered at Origin

Here we will see how an ellipse equation in general form, centered at (h,k) is plotted on a graph. Ellipse having equation in general form is px² + qy² + cx + dy + e = 0; where (p,q)>0

Now, let us learn to plot an ellipse on a graph using an equation as in the above form.

Let’s take the equation 9x² + 4y² + 18x – 16y -11 = 0  and identify whether it is a horizontal or vertical ellipse. We will also label the center, vertices, co-vertices, and foci.

First, let us rewrite the equation in the standard form.

9x² + 4y² + 18x – 16y -11  = 0

(p=9, q=4, c=18, d = -16, e = -11)

or, 9x² + 18x + 4y² + 16y = 11

or, 9(x² + 2x) + 4(y²-4x) = 11

Converting the expression to (a±b)²

we can write the expression as following:

9(x² + 2x + __) + 4(y²-4x + __) = 11

To get the values of __ in above expression,

for x, we divide the co-efficient of x (i.e. 2) by 2. We get 2/2 = 1

Similarly for y, we divide the co-efficient of y (i.e. 4) by 2. We get  4/2 = 2

So we write the expression as :

9(x² + 2x + 1) + 4(y²-4x + 4) = 11 + 9 + 16 (equating the two sides, we need to add 16 + 9 on both sides)

or, 9(x+1)² + 4(y-2)² = 36

or, [9(x+1)² + 4(y-2)²]/36 = 1

or, [(x+1)²]/4+ [(y-2)²]/9 = 1 (we get this expression as the equation of ellipse in standard form)

From this given equation, , it is clear that the ellipse is vertical, seeing the positions of a² and b² as in the denominators. So, the foci & vertices must be on the y-axis.

∴ center (h, k) : h = -1, k = 2 Hence, center = (-1, 2)

a² = 9; a = 3, b² = 4; b = 2

so, the vertices are (-1, 5) & (-1, -1)

co-vertices are (1,2) & (-3,2)

F² = a² – b² =  9-4 = 5 , F = √5 ≈ 2.236

Foci = (-1, 2± √5) = (-1,4.236) & (-1, – 0.236)

Now let’s graph the ellipse.

How to Graph an Ellipse in Standard Form: Centered at Origin

Here we will see how an ellipse equation in standard form, centered at the origin is plotted on a graph.

Let’s take the equation x²/9 + y²/1= 1 and identify whether it is a horizontal or vertical ellipse. We will also label the center, vertices, co-vertices, and foci.

Comparing the given equation in the question, with the standard equation of an ellipse x²/a² + y²/b² = 1centered at (0,0),

From the equation, we understand, the foci & vertices must be on the x-axis  hence it is a horizontal ellipse, seeing the positions of a² and b² as in the denominators. So, the foci & vertices must be on the x-axis.

∴Center = (0,0)

a= 3, b = 1

Vertices: (±a, 0) = (3,0) & (-3,0)

Co-vertices: (0, ±b) = (0,1) & (0,-1)

Equation of focal length is F² = a² – b²  = 9-1 = 8, F = √8 ≈ 2.83

∴ Foci = (±√8, 0) = (2.83,0) & (-2.83,0)

Now let’s graph the ellipse.

How to Graph an Ellipse in General Form: Centered at Origin

Here we will see how an ellipse equation in general form, centered at the origin is plotted on a graph.

Ellipses centered at zero are represented in general form as px² + qy² = e = 0;where (p,q)>0

Now, let us learn to plot an ellipse on a graph using an equation as in the above form.

Let’s take the equation x² + 16y² -16= 0 and identify whether it is a horizontal or vertical ellipse. We will also label the center, vertices, co-vertices, and foci.

Rewriting the equation into standard form as follows:

x² + 16y² -16= 0

or, x² + 16y² = 16

or, x²/16+ 16y²/16 = 16/16 (dividing both sides by 16)

or, x²/16 + y²/1 = 1 (standard form)

From the equation, we understand, the foci & vertices must be on the x-axis  hence it is a horizontal ellipse, seeing the positions of a² and b² as in the denominators. So, the foci & vertices must be on the x-axis.

∴, center (h,k) = (0,0), a = 4, b = 1,

Vertices: (±a, 0) = (4,0) & (-4,0)

Co-vertices: (0, ±b) = (0,1) & (0,-1)

F² = 16-1 = 15, F = √15 ≈ 3.87 ≈ 3.8

∴ Foci = (±√8, 0) = (3.8,0) & (-3.8,0)

Now let’s graph the ellipse.