What is the equation of the ellipse having foci (±2, 0) and the eccentricity \(\frac{1}{4}?\)
Finding the Ellipse Equation from Foci and Eccentricity
The problem asks for the equation of an ellipse given its foci and eccentricity. We are given that the foci are located at \((\pm 2, 0)\) and the eccentricity \(e = \frac{1}{4}\).
The location of the foci along the x-axis indicates that the major axis of the ellipse lies on the x-axis. This means the standard equation of the ellipse centered at the origin is of the form:
\(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)
where \(a\) is the semi-major axis length and \(b\) is the semi-minor axis length.
Using Foci Information to Find 'c'
For an ellipse with foci at \((\pm c, 0)\), the distance from the center to each focus is \(c\). In this case, the foci are given as \((\pm 2, 0)\). Therefore, we have:
\(c = 2\)
Using Eccentricity to Find 'a'
The eccentricity \(e\) of an ellipse is defined as the ratio of the distance from the center to the focus (\(c\)) to the length of the semi-major axis (\(a\)). The formula is:
\(e = \frac{c}{a}\)
We are given \(e = \frac{1}{4}\) and we found \(c = 2\). We can substitute these values into the formula to find \(a\):
\(\frac{1}{4} = \frac{2}{a}\)
Solving for \(a\):
\(1 \times a = 4 \times 2\)
\(a = 8\)
Now we can find \(a^2\):
\(a^2 = 8^2 = 64\)
Finding 'b\(\textsuperscript{2}\)' using 'a' and 'c'
For an ellipse with the major axis on the x-axis, the relationship between \(a\), \(b\), and \(c\) is given by:
\(c^2 = a^2 - b^2\)
We have \(c = 2\) and \(a = 8\). First, let's find \(c^2\):
\(c^2 = 2^2 = 4\)
Now substitute \(a^2 = 64\) and \(c^2 = 4\) into the relationship formula:
\(4 = 64 - b^2\)
Now, solve for \(b^2\):
\(b^2 = 64 - 4\)
\(b^2 = 60\)
Writing the Ellipse Equation
Now that we have \(a^2 = 64\) and \(b^2 = 60\), we can substitute these values into the standard equation of the ellipse centered at the origin with the major axis on the x-axis:
\(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)
Substituting the values:
\(\frac{x^2}{64} + \frac{y^2}{60} = 1\)
This is the equation of the ellipse with the given foci and eccentricity.
| Property | Value |
|---|---|
| Foci | \((\pm 2, 0)\) |
| Eccentricity (\(e\)) | \(\frac{1}{4}\) |
| Distance from center to focus (\(c\)) | \(2\) |
| Semi-major axis (\(a\)) | \(8\) |
| Semi-minor axis squared (\(b^2\)) | \(60\) |
| Semi-major axis squared (\(a^2\)) | \(64\) |
Revision Table: Key Concepts for Ellipse Equation
- Standard equation for ellipse centered at origin (horizontal major axis): \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
- Standard equation for ellipse centered at origin (vertical major axis): \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\).
- Foci for horizontal major axis ellipse: \((\pm c, 0)\).
- Foci for vertical major axis ellipse: \((0, \pm c)\).
- Relationship between \(a\), \(b\), and \(c\) for horizontal major axis ellipse: \(c^2 = a^2 - b^2\) (where \(a > b\)).
- Relationship between \(a\), \(b\), and \(c\) for vertical major axis ellipse: \(c^2 = a^2 - b^2\) (where \(a > b\)).
- Eccentricity formula: \(e = \frac{c}{a}\). Eccentricity is always between 0 and 1 for an ellipse (\(0 \le e < 1\)).
Additional Information: Properties of Ellipses
An ellipse is a type of conic section formed by intersecting a cone with a plane at an angle to its base. Key properties include:
- Definition: An ellipse is the set of all points in a plane such that the sum of the distances from two fixed points (the foci) is constant. This constant sum is equal to \(2a\), the length of the major axis.
- Center: The midpoint of the line segment connecting the foci. For the standard equations discussed, the center is at the origin \((0,0)\).
- Vertices: The endpoints of the major axis. For a horizontal ellipse centered at the origin, the vertices are \((\pm a, 0)\).
- Co-vertices: The endpoints of the minor axis. For a horizontal ellipse centered at the origin, the co-vertices are \((0, \pm b)\).
- Major Axis: The longest diameter of the ellipse, passing through the foci and vertices. Its length is \(2a\).
- Minor Axis: The shortest diameter of the ellipse, perpendicular to the major axis and passing through the center and co-vertices. Its length is \(2b\).
- Eccentricity: A measure of how "stretched out" the ellipse is. An eccentricity close to 0 means the ellipse is nearly circular. An eccentricity close to 1 means the ellipse is very elongated.
Understanding these properties is crucial for solving various problems involving ellipses.
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