Ellipses

Exercises 25 and 26 give information about the foci and vertices of ellipses centered at the origin of the xy−plane. In each case, find the ellipse's standard−form equation from the given information.

Foci: ( ±√2, 0) Vertices: (±2,0)

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Identify the orientation of the ellipse based on the given points. Since the foci and vertices lie on the x-axis (points have the form (±x, 0)), the major axis is horizontal.

Recall the standard form of the ellipse equation centered at the origin with a horizontal major axis: \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\), where \(a\) is the semi-major axis length and \(b\) is the semi-minor axis length.

Determine \(a\) from the vertices. The vertices are at \((\pm 2, 0)\), so \(a = 2\) and therefore \(a^{2} = 4\).

Determine \(c\) from the foci. The foci are at \((\pm \sqrt{2}, 0)\), so \(c = \sqrt{2}\) and therefore \(c^{2} = 2\).

Use the relationship between \(a\), \(b\), and \(c\) for ellipses: \(c^{2} = a^{2} - b^{2}\). Substitute the known values to solve for \(b^{2}\), then write the standard form equation \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\).

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주요 개념

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Standard Form of an Ellipse Centered at the Origin

An ellipse centered at the origin has the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) if the major axis is along the x-axis, where \(a\) is the semi-major axis length and \(b\) is the semi-minor axis length. Knowing the vertices and foci helps determine \(a\), \(b\), and \(c\) (distance from center to foci).

Relationship Between Vertices, Foci, and Axes Lengths

For an ellipse, the distance from the center to each vertex is \(a\), and the distance to each focus is \(c\). These satisfy the equation \(c^2 = a^2 - b^2\). Given vertices and foci, you can find \(a\), \(c\), and then calculate \(b\) to write the ellipse equation.

Identifying the Orientation of the Ellipse

The orientation of the ellipse (horizontal or vertical major axis) is determined by the coordinates of the vertices and foci. Since both vertices and foci lie on the x-axis (\(y=0\)), the major axis is horizontal, so \(x^2\) is over \(a^2\) in the standard form.

EllipsesExercises 25 and 26 give information about the foci and vertices of ellipses centered at the origin of the xy−plane. In each case, find the ellipse's standard−form equation from the given information.Foci: ( ±√2, 0) Vertices: (±2,0)