Find the standard form of the equation of each ellipse satisfying the given conditions. Foci: (-2, 0), (2, 0); y-intercepts: -3 and 3

Verified step by step guidance

1. Identify the center of the ellipse. Since the foci are at (-2, 0) and (2, 0), the center is the midpoint of the foci, which is at (0, 0).
2. Determine the orientation of the ellipse. The foci lie on the x-axis, so the major axis is horizontal.
3. Calculate the distance between the center and each focus, denoted as \(c\). Here, \(c = 2\) because the foci are at \(\pm 2\) on the x-axis.
4. Use the y-intercepts to find the length of the minor axis. The y-intercepts are at \(y = \pm 3\), so the minor axis length is \$2b = 6\(, giving \)b = 3$.
5. Apply the relationship between \(a\), \(b\), and \(c\) for ellipses: \(c^2 = a^2 - b^2\). Use this to solve for \(a^2\), then write the standard form equation of the ellipse centered at the origin with a horizontal major axis: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

Key Concepts

Standard Form of an Ellipse

The standard form of an ellipse equation depends on the orientation of its major axis. For a horizontal major axis centered at the origin, it is (x²/a²) + (y²/b²) = 1, where 'a' is the semi-major axis and 'b' is the semi-minor axis. Understanding this form helps in writing the ellipse equation once 'a' and 'b' are known.

Foci and Relationship to Axes

The foci of an ellipse are two fixed points used to define the curve. The distance from the center to each focus is 'c', and it relates to the axes by the equation c² = a² - b² for horizontal ellipses. Knowing the foci helps determine 'c', which is essential to find 'a' and 'b'.

Intercepts and Their Role in Determining Axis Lengths

The y-intercepts of an ellipse give the points where the ellipse crosses the y-axis, indicating the length of the minor axis. Since the ellipse crosses the y-axis at ±b, the intercepts help find the value of 'b'. This information, combined with 'c', allows solving for 'a' and writing the equation.