Precalculus Notes: The Ellipse
📝 Definition and Key Components
An ellipse is the set of all points in a plane where the sum of the distances from two
fixed points, called the foci (plural of focus), is constant.
● Center: The midpoint of the major axis and the minor axis, denoted by (h,k).
● Major Axis: The longest diameter of the ellipse. Its length is 2a.
● Minor Axis: The shortest diameter of the ellipse. Its length is 2b.
● Vertices: The endpoints of the major axis.
● Foci: The two fixed points used in the definition. The distance from the center to
each focus is c.
The relationship between a, b, and c is given by the equation:
c²=a²−b²
This equation is crucial for finding the foci once you have the values of a and b.
📐 The Equations of an Ellipse
The standard form of an ellipse's equation tells you its center, major axis, and minor
axis.
Horizontal Ellipse
The major axis is horizontal. This occurs when the larger denominator is under the x2
term.
a²(x−h)²+b²(y−k)2²=1
● Center: (h,k)
● Vertices: (h±a,k)
● Foci: (h±c,k)
Vertical Ellipse
The major axis is vertical. This occurs when the larger denominator is under the y2
term.
b²(x−h)²+a²(y−k)²=1
● Center: (h,k)
● Vertices: (h,k±a)
● Foci: (h,k±c)
Key Tip: The value of a² is always the larger of the two denominators. The orientation
(horizontal or vertical) is determined by whether a2 is under the x term or the y term.
🔄 Example
Let's find the center, vertices, and foci of the ellipse given by the equation:
9(x+1)2+25(y−3)2=1
1. Identify the Center: By comparing the equation to the standard form, we can
see that h=−1 and k=3. The center is (−1,3).
2. Determine the Orientation: The larger denominator is 25, which is under the y2
term. Therefore, the major axis is vertical.
3. Find a and b: Since a²=25, we have a=5. Since b²=9, we have b=3.
4. Find the Vertices: The vertices are (h,k±a). So, we have (−1,3±5), which gives us
(−1,8) and (−1,−2).
5. Find the Foci: First, find c using c²=a²−b².
○ c²=25−9=16
○ c=16=4
○ The foci are (h,k±c). So, we have (−1,3±4), which gives us (−1,7) and (−1,−1).
🧠 Converting from General to Standard Form
Similar to circles, an ellipse can be given in general form, such as
4x2+9y2−16x+18y−11=0. To find the center, vertices, and foci, you must use the
method of completing the square to convert it back to standard form.
