Ellipses and Eccentricity

Exercises 9–12 give the foci or vertices and the eccentricities of ellipses centered at the origin of the xy-plane. In each case, find the ellipse’s standard-form equation in Cartesian coordinates.

Vertices: (±10,0)
Eccentricity: 0.24

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Identify the orientation of the ellipse based on the vertices. Since the vertices are at (±10, 0), the major axis is along the x-axis, so the ellipse equation will be of the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

Determine the value of \(a\), which is the distance from the center to each vertex. Here, \(a = 10\), so \(a^2 = 100\).

Recall the relationship between the eccentricity \(e\), the semi-major axis \(a\), and the distance to each focus \(c\): \(e = \frac{c}{a}\). Use the given eccentricity \(e = 0.24\) to find \(c = e \times a\).

Use the relationship between \(a\), \(b\), and \(c\) for ellipses: \(c^2 = a^2 - b^2\). Rearrange this to solve for \(b^2\): \(b^2 = a^2 - c^2\).

Substitute the values of \(a^2\) and \(b^2\) into the standard form equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) to write the equation of the ellipse.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Standard Form of an Ellipse

The standard form of an ellipse centered at the origin is either (x²/a²) + (y²/b²) = 1 for a horizontal major axis or (x²/b²) + (y²/a²) = 1 for a vertical major axis, where a is the semi-major axis length and b is the semi-minor axis length. Knowing the vertices helps determine the value of a.

Eccentricity of an Ellipse

Eccentricity (e) measures how elongated an ellipse is, defined as e = c/a, where c is the distance from the center to each focus and a is the semi-major axis length. It ranges from 0 (circle) to 1 (parabola). Given e and a, you can find c, which helps locate the foci.

Relationship Between a, b, and c in an Ellipse

In an ellipse, the relationship c² = a² - b² connects the semi-major axis (a), semi-minor axis (b), and focal distance (c). This formula allows you to find b once a and c are known, completing the parameters needed to write the ellipse's equation.

Ellipses and EccentricityExercises 9–12 give the foci or vertices and the eccentricities of ellipses centered at the origin of the xy-plane. In each case, find the ellipse’s standard-form equation in Cartesian coordinates.Vertices: (±10,0)Eccentricity: 0.24