Chapter 8.4, Problem 2E

Solution

To explain : The method to find the foci of an ellipse when the coordinates of its vertices and co-vertices are given.

The foci of an ellipse can be found using the formula $c^2=a^2+b^2$ , where $c$ is the foci, $a$ is the vertex, and $b$ is the co-vertex.

Given information :

An ellipse and the coordinates of its vertices and co-vertices.

Explanation :

The vertex, co-vertex, and focus of an ellipse with horizontal major axis is given by $\left(\pm a,0\right),\left(0,\pm b\right),\text{ and }\left(\pm c,0\right)$ respectively.

The vertex, co-vertex, and focus of an ellipse with vertical major axis is given by $\left(0,\pm a\right),\left(\pm b,0\right),\text{ and }\left(0,\pm c\right)$ respectively.

The formula for foci is $c^2=a^2+b^2$ .

Therefore, the foci of an ellipse can be found using the formula $c^2=a^2+b^2$ , where $c$ is the foci, $a$ is the vertex, and $b$ is the co-vertex.