The problem asks for the $xy$-equation of a vertical ellipse centered at $(0, 0)$ with a major diameter of 8 and a minor diameter of 6.

1. Problem Description

The problem asks for the $xy$-equation of a vertical ellipse centered at $(0, 0)$ with a major diameter of 8 and a minor diameter of

6.

2. Solution Steps

The general equation of an ellipse centered at $(0, 0)$ is given by:

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

where $a$ and $b$ are the semi-minor and semi-major axes, respectively, when $b > a$.

The major diameter is 8, which means the semi-major axis is $b = \frac{8}{2} = 4$.

The minor diameter is 6, which means the semi-minor axis is $a = \frac{6}{2} = 3$.

Since it is a vertical ellipse, the major axis is along the y-axis, so $b$ corresponds to the $y$ term and $a$ corresponds to the $x$ term.

Therefore, the equation of the ellipse is:

$\frac{x^2}{3^2} + \frac{y^2}{4^2} = 1$

$\frac{x^2}{9} + \frac{y^2}{16} = 1$

3. Final Answer

The $xy$-equation of the vertical ellipse is $\frac{x^2}{9} + \frac{y^2}{16} = 1$.