Find the standard form of the equation of an ellipse with vertices at (0, -6) and (0, 6), passing through (2, 4).

Verified step by step guidance

1. Identify the center of the ellipse by finding the midpoint of the vertices. Since the vertices are at (0, -6) and (0, 6), the center is at (0, 0).
2. Determine the orientation of the ellipse. The vertices lie on the y-axis, so the major axis is vertical. This means the standard form of the ellipse equation is \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \), where \(a\) is the distance from the center to a vertex along the y-axis.
3. Calculate \(a\), the distance from the center to a vertex. Since the vertices are at (0, ±6), \(a = 6\). So, \(a^2 = 36\).
4. Use the point (2, 4) that lies on the ellipse to find \(b^2\). Substitute \(x = 2\), \(y = 4\), and \(a^2 = 36\) into the ellipse equation: \( \frac{2^2}{b^2} + \frac{4^2}{36} = 1 \).
5. Solve the equation from step 4 for \(b^2\) by isolating \( \frac{4}{b^2} \), then multiply both sides by \(b^2\) and solve for \(b^2\). This will give you the value needed to write the standard form of the ellipse.

Key Concepts

Standard Form of an Ellipse

The standard form of an ellipse equation depends on the orientation of its major axis. For a vertical major axis centered at the origin, the equation is (x^2 / b^2) + (y^2 / a^2) = 1, where 'a' is the distance from the center to a vertex along the y-axis, and 'b' is the distance along the x-axis. Understanding this form helps in identifying the ellipse's shape and size.

Vertices of an Ellipse

Vertices are the points on the ellipse farthest from the center along the major axis. Given vertices at (0, -6) and (0, 6), the center is at the origin (0,0), and the length of the major axis is 12, so a = 6. Knowing the vertices allows determination of 'a' in the ellipse equation.

Using a Point to Find the Ellipse Parameters

Substituting a known point on the ellipse into the standard form equation allows solving for the unknown parameter 'b'. Here, the point (2, 4) lies on the ellipse, so plugging in x=2 and y=4 helps find 'b', completing the equation. This step is essential to fully define the ellipse.