1. The problem states that the vertices of the ellipse are at coordinates $(0, a)$ and $(0, -a)$, and the foci are at $(0, c)$ and $(0, -c)$ with the relationship $$c^2 = a^2 - b^2.$$ 2. This describes a vertical ellipse centered at the origin. The standard form of the ellipse equation is $$\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1,$$ where $a > b > 0$. Here, $a$ is the distance from the center to each vertex along the y-axis, and $b$ is the distance from the center to the ellipse along the x-axis. 3. The foci are located at $(0, c)$ and $(0, -c)$, where $c$ is the focal distance from the center. The relationship between $a$, $b$, and $c$ is given by $$c^2 = a^2 - b^2.$$ This means the foci lie inside the ellipse along the y-axis. 4. To find $c$, you can rearrange the formula: $$c = \sqrt{a^2 - b^2}.$$ This is important because it tells you how far the foci are from the center. 5. In summary, the ellipse has vertices at $(0, a)$ and $(0, -a)$, foci at $(0, c)$ and $(0, -c)$, and the equation $$\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$$ with $$c^2 = a^2 - b^2.$$ This fully describes the ellipse's geometry.