Finding the Equation of an Ellipse Given Its Axes and Vertices

Understanding how to find the equation of an ellipse given its axes and vertices is a fundamental skill in analytic geometry. This process involves several steps, including determining the center, the lengths of the semi-major and semi-minor axes, and finally, formulating the equation. In this article, we will walk through these steps with examples and explanations.

Step 1: Identifying the Center and Axes

To find the equation of an ellipse given its vertices and axes, the first step is to identify the center and the orientation of the ellipse. The center of the ellipse can often be determined by the midpoint of the vertices. If the vertices are given, and they lie on the coordinate axes, you can easily infer the orientation of the ellipse.

Example

Suppose we are given the vertices of an ellipse as ( (0, 5) ) and ( (0, -5) ). These vertices lie on the y-axis, indicating that the major axis is vertical and the center of the ellipse is at the origin ( (0,0) ).

Step 2: Determining the Lengths of the Semi-Axes

The next step is to determine the lengths of the semi-major and semi-minor axes. The semi-major axis (a) is the distance from the center to the longer vertex, and the semi-minor axis (b) is the distance from the center to the shorter vertex.

Example

Continuing with our example, the distance from the center ( (0,0) ) to the vertex ( (0, 5) ) is 5 units. Therefore, the length of the semi-major axis ( a 5 ). If we have another vertex on the x-axis, for instance ( (4, 0) ), the distance from the center to this vertex is 4 units, indicating that the semi-minor axis ( b 4 ).

The relationship between the semi-major axis ( a ), the semi-minor axis ( b ), and the distance to the foci ( c ) is given by the formula:

c^2 a^2 - b^2

Determining the Shorter Axis

If the given vertices do not directly indicate the shorter axis, you will need to use the vertices and the foci to determine the length of the semi-minor axis. The foci of the ellipse are located at ( (pm c, 0) ) for a horizontal ellipse and at ( (0, pm c) ) for a vertical ellipse.

Formulating the Equation

Once you have the values of ( a ) and ( b ), you can write the equation of the ellipse in its standard form:

[ frac{x^2}{a^2} frac{y^2}{b^2} 1 quad text{if the major axis is horizontal} ] [ frac{x^2}{b^2} frac{y^2}{a^2} 1 quad text{if the major axis is vertical} ]

Example Calculation

Using our example, if the center is at the origin and the semi-major axis ( a 5 ) and the semi-minor axis ( b 4 ), the equation of the ellipse is:

[ frac{x^2}{5^2} frac{y^2}{4^2} 1 ]

This simplifies to:

[ frac{x^2}{25} frac{y^2}{16} 1 ]

Similarly, if the semi-major axis was ( b 5 ) and the semi-minor axis ( a 4 ), the equation would be:

[ frac{x^2}{4^2} frac{y^2}{5^2} 1 ]

This simplifies to:

[ frac{x^2}{16} frac{y^2}{25} 1 ]

Conclusion

By following these steps, you can find the equation of an ellipse given its vertices and axes. Whether the major axis is horizontal or vertical, the method remains consistent. Understanding these concepts will help you solve more complex problems involving ellipses in analytic geometry.

FAQ

Q: What if the ellipse is not centered at the origin?

A: If the center of the ellipse is not the origin, you need to include the center coordinates ((h, k)) in the standard form of the ellipse:

[ frac{(x-h)^2}{a^2} frac{(y-k)^2}{b^2} 1 ]

This will shift the ellipse to the coordinates ((h, k)).

Q: What if the vertices are not given?

A: If only the foci and other points are given, you can still determine the lengths of the semi-major and semi-minor axes using the relationships between these points and the foci.

Q: How do I handle ellipses with rotations?

A: For ellipses rotated about the origin, the equation becomes more complex and includes trigonometric terms. However, the basics of finding (a) and (b) still apply; you just need to account for the rotation in your calculations.