There are four types of conic sections, one of which is an ellipse. An ellipse can sometimes resemble a circle or will be more long and narrow, almost oval shaped. The ellipse is a set of all points in a single plane where the total distance from two fixed points is constant.
The equation of the ellipse in standard form that is symmetric with both axes with a center of (0, 0) is given by (x 2)/a 2+ (y 2)/ b 2 = 1 where a > 0 and b > 0. The intercepts of the graph are (a , 0), (-a, 0), (0, b) and (0, -b). In an ellipse that is elongated horizontally, a > b. This is easy to remember because the larger number will be under the x2 and the elongation is along the x- axis. If a 2 and the elongation is along the y- axis. The foci (plural of focus) are at (c, 0) and (-c, 0), where c2 = a2 – b2 . The vertices are the endpoints of each axis. The line segment joining the vertices of the elongated side is called the major axis and the line segment joining the vertices of the shorter side is called the minor axis. The following examples will illustrate all of the points above about the ellipse centered at (0, 0).
Graph: (x 2)/25 + (y 2)/9 = 1.
We will first plot the intercepts by solving for a and b. From the standard equation of the ellipse, we know that a2 = 25 and b2 = 9. Therefore a = 5 and b = 3. The intercepts are then (5, 0), (-5, 0), (0, 3) and (0, -3). Since a > b we know that major axis is the x- axis and the minor axis is the y- axis. We calculate the value of c from c2 = b2 – a2 to plot the foci. Therefore c2 = 25 – 9 = 16, so c = 4. The foci are at (4, 0) and (-4, 0). We can draw the ellipse through the 4 points or we can add more points by substituting values in for x or y. For practical purposes, there is no need to add extra points
Example: Graph 36×2 + y2 = 36.
In this example, we must first notice that the equation is not in standard form. To get the equation in standard form, we must divide the entire equation by 36. This will set the equation equal to 1.
(36 x 2 + y 2 = 36)/36 = x 2 + (y 2)/36 = 1.
Now we can plot the intercepts by solving for a and b. We know that a2 = 1 and b2 = 36, therefore a = 1 and b = 6. The intercepts are (1, 0), (-1, 0), (0, 6) and (0, -6). Since a 2 = b2 – a2 to plot the foci. Therefore c2 = 36 -1 = 35, therefore c ≈ 5.9 and the foci are (0, 5.9) and (0, -5.9).
If the equation is in the form (x 2)/b 2 + (y 2)/a 2 = 1, then the major axis is the y- axis. Oftentimes an ellipse will not be centered at the origin. The equation of an ellipse in standard form with the center at (h, k) is given by
( x – h ) 2/a 2 + ( y – k ) 2 /b 2 = 1, where a > 0 and b > 0.
Example: Graph (x – 2) 2/ 9 + ( y – 4) 2/ 16 = 1.
We will first plot the vertices by solving for a and b. Since a2 = 9 and b2 = 16, we know that a = 3 and b = 4. The vertices are 3 units in both directions from the x- coordinate of the center and 4 units in both directions from the y- coordinate of the center. The center is at (2, 4), therefore the vertices are (5, 4), (-1, 4), (2, 8) and (2, 0). We know that the y- axis is the major axis and the x- axis is the minor axis. We calculate the value of c from c2 = a2 – b2 to plot the foci. Therefore c2 = 16 – 9 = 7, so c ≈ 2.65. Remember that the foci are along the major axis, so we move c units in either direction from the center along the y- axis. The foci are (2, 4 + 2.65) and (2, 4 – 2.65) or (2, 6.65) and (2, 1.35).
Sometimes information about an ellipse is given and you must write the equation.
Example: Find the equation of an ellipse in standard form with foci (6, -2) and (-2, -2) and major axis of length 10.
Since the y- coordinates of the foci are the same, we know the major axis is the x- axis. Recall that the foci are the same distance from the center. Since the foci are 8 units apart (6 – (-2) = 8), we move four units from either focus point to get the center (6 -4, -2) = (2, -2) or (-2 + 4, -2) = (2, -2). Therefore c = 4 and the center of the ellipse is at (2, -2). Since x is the major axis and the length of the axis is 10, we know that a = 5. Knowing a and c enables us to find b. We know that c2 = a2 – b2, therefore 42 = 52 – b2 and b = 3.
From the center (2, -2) we can find the vertices using a = 5 and b = 3. Find the vertices by moving 5 units in both directions along the x- axis from the center and 3 units in both directions along the y- axis from the center. Therefore the vertices are (7, -2), (-3, -2), (2, 1) and (2, -5). Substitute (2, -2) for (h, k), 5 for a and 3 for b to get the equation as ( x – 2) 2/ 25 + ( y + 2) 2/ 9 = 1.
This guide on the ellipse should assist any student having difficulty understanding how to graph and write the equation of an ellipse.