Question

Find the equation of the ellipse defined by the given information. Sketch the ellipse. Foci: (±2,0)\(;\) vertices: (±3,0)

Short answer

The equation of the ellipse is \(\frac{x^2}{9} + \frac{y^2}{5} = 1\).

Step-by-step solution

Understand the standard form of an ellipse equation

The standard form of the equation for an ellipse centered at the origin with a horizontal major axis is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here, \(a\) is the distance from the center to a vertex on the major axis, and \(b\) is the distance from the center to a vertex on the minor axis.

Identify values of a, b, and c

Given that the vertices are (±3,0), this means \(a=3\). The foci are (±2,0), which indicates \(c=2\). In an ellipse, \(c^2 = a^2 - b^2\).

Calculate b using the ellipse formula

Substitute the given values into the equation \(c^2 = a^2 - b^2\):\[2^2 = 3^2 - b^2\]\[4 = 9 - b^2\]Solve for \(b^2\):\[b^2 = 9 - 4 = 5\]Thus, \(b = \sqrt{5}\).

Write down the equation of the ellipse

Now that we know \(a = 3\) and \(b = \sqrt{5}\), the equation of the ellipse can be written as:\[\frac{x^2}{3^2} + \frac{y^2}{(\sqrt{5})^2} = 1\]\[\frac{x^2}{9} + \frac{y^2}{5} = 1\]

Sketch the ellipse

Draw an ellipse centered at the origin (0,0). The major axis is horizontal with endpoints at the vertices (±3,0). The minor axis is vertical with endpoints (0,±\(\sqrt{5}\)). The foci are at (±2,0).

Key concepts

Foci of an Ellipse

Foci are unique points located inside an ellipse. They play a crucial role in defining its shape. For any point on the ellipse, the sum of distances to the two foci is a constant, which is equal to the length of the major axis.
In our example, the foci are given as (±2,0). This means the foci are symmetrically placed along the x-axis at two units from the center, which is at the origin (0,0). These foci determine how "stretched" or "flattened" the ellipse appears.
The property of the foci ensures that if you pick any point on the ellipse, add up the distances from this point to both foci, you will always get the same result. This constant sum is crucial for calculating various properties of the ellipse.

Major and Minor Axes

The major and minor axes are the main reference lines of an ellipse, defining its orientation and proportions.
- **Major Axis**: This is the longest diameter of the ellipse. It's the line segment that includes both the center and the foci of the ellipse. In our case, the major axis is horizontal because the vertices are located at (±3,0). Therefore, the distance from the center to a vertex—denoted by 'a'—is 3.- **Minor Axis**: This segment is perpendicular to the major axis and passes through the center of the ellipse. Its endpoints are above and below the center. The length of the minor axis is determined using 'b', which is found via the equation: \[c^2 = a^2 - b^2\] In this example, since we use values 'a' = 3 and 'c' = 2, we calculate 'b' as \(b = \sqrt{5}\), giving the minor axis length \(2\sqrt{5}\). The minor axis is shorter than the major axis, and it stretches along the y-axis with endpoints (0, ±\(\sqrt{5}\)).

Standard Form of Ellipse Equation

The standard form of an ellipse equation allows us to express how an ellipse is structured in terms of its axes.
The general standard equation for an ellipse centered at the origin is:
\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]
Here, 'a' and 'b' represent the semi-major and semi-minor axes, respectively. 'a' is larger when the ellipse is stretched horizontally, while 'b' takes the larger value if stretched vertically.
In our exercise, the ellipse is horizontal, so 'a' = 3 and 'b' = \(\sqrt{5}\). Substituting these values, we simplify the equation to:
\[\frac{x^2}{9} + \frac{y^2}{5} = 1\]
This equation is an essential tool for graphically representing and calculating the various properties of the ellipse, ensuring a complete understanding of its dimensions and orientation.