In Exercises 19–24, find the standard form of the equation of each ellipse and give the location of its foci.

Verified step by step guidance

1. Identify the center of the ellipse, which is at the origin (0,0).
2. Determine the lengths of the semi-major axis and semi-minor axis. The semi-major axis is 10 units (from (0,10) to (0,-10)), and the semi-minor axis is 6 units (from (-6,0) to (6,0)).
3. Write the standard form of the equation of the ellipse: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \(a\) is the length of the semi-major axis and \(b\) is the length of the semi-minor axis.
4. Substitute the values of \(a\) and \(b\) into the equation: \( \frac{x^2}{6^2} + \frac{y^2}{10^2} = 1 \).
5. To find the foci, use the formula \(c = \sqrt{a^2 - b^2}\). Calculate \(c\) and then determine the coordinates of the foci, which will be at (0, c) and (0, -c).

Key Concepts

Standard Form of an Ellipse

The standard form of the equation of an ellipse is given by (x-h)²/a² + (y-k)²/b² = 1, where (h, k) is the center, a is the semi-major axis, and b is the semi-minor axis. This form helps identify the orientation of the ellipse (horizontal or vertical) and its dimensions based on the values of a and b.

Foci of an Ellipse

The foci of an ellipse are two fixed points located along the major axis, which are crucial for defining the shape of the ellipse. The distance from the center to each focus is denoted as c, where c² = a² - b². The foci are essential for understanding the properties of the ellipse, such as its eccentricity.

Vertices of an Ellipse

The vertices of an ellipse are the points where the ellipse intersects its major and minor axes. For a horizontally oriented ellipse, the vertices are located at (h ± a, k) and for a vertically oriented ellipse at (h, k ± b). Identifying the vertices is important for sketching the ellipse and understanding its dimensions.