The equation of an ellipse with its centre at the origin has one of two forms:

## Horizontal Ellipse

**Foci:**

** **

## Vertical Ellipse:

** Foci:**

For a horizontal Major Axis and C(0,0): major axis = 2a and minor axis = 2b

For a Horizontal Major Axis and C(h,k):

For a Vertical Major Axis and C(0,0), major axis = 2a and minor axis = 2b:

For a vertical Major Axis and C(h,k):

From the standard equations as explained above, we can make the following observations:

- An ellipse is symmetric with respect to the major and minor axis. If (x, y) is a point on the ellipse, (-x, y), (x, -y) and (-x, -y) also exist on the ellipse.
- The foci are present on the major axis which can be deduced by finding the intercepts on the axes of symmetry. This means that the major axis is along the x-axis when the coefficient x
^{2}has the bigger denominator. The major axis is along the y-axis when the coefficient of y^{2}has the bigger denominator.

Steps for writing the equation of the ellipse in standard form:

- Complete the square for both the x-terms and y-terms and move the constant to the other side of the equation
- Divide all terms by the constant

**Example 1:**

Steps for graphing the ellipse:

- Put equation in standard form
- Graph the centre (h, k)
- Graph the foci (look at the equation to determine your direction)
- Graph a units and –units from the centre to get the endpoints (horizontally if under x, vertically if under y)
- Connect the endpoints

* Example 2:* Write the following equation in standard form, then graph it:

* Example 3:* Find the equation of the ellipse with foci at (8,0) and (-8,0) and a vertex at (12,0).

- First, place these points on axes. The F and F’ are the foci.
- Since the vertex is on the horizontal axis, the ellipse will be of the form:
- The values of
*a*and b need to be determined. - Because the foci are at 8 and -8,
*c = 8.*Also, the vertex is at (12,0), hence*a = 12.*Relating these values to the standard form for an ellipse whose centre is at the origin and whose major axis is horizontal, the equivalence c^{2}= a^{2 }– b^{2 }applies. Solve for*b*where^{2 } - The value of
*a*is 12, and a^{2 }is 144. - So the equation of the ellipse is: