The given problem asks to find the standard equation of an ellipse, where:

• The ends of the major axis are $(0, \pm 9)$.
• The ends of the minor axis are $(\pm 3, 0)$.

From this information, we can conclude the following:

1. Center of the Ellipse: Since the axes are centered on the origin and the given points are symmetric about the origin, the center of the ellipse is $(0, 0)$.

2. Length of the Major Axis: The major axis runs vertically (because the $y$-coordinates vary more). The distance from the center to the endpoints of the major axis is 9. Therefore, the length of the semi-major axis $a = 9$.

3. Length of the Minor Axis: The minor axis runs horizontally (since the $x$-coordinates vary in the minor axis points). The distance from the center to the endpoints of the minor axis is 3. Therefore, the length of the semi-minor axis $b = 3$.

Standard Equation of an Ellipse:

The standard form of the equation for an ellipse with a vertical major axis is:

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

Substituting $a = 9$ and $b = 3$:

[ \frac{x^2}{3^2} + \frac{y^2}{9^2} = 1 \