To determine the standard equation of an ellipse, we need the center, the values for the major and minor axes, and the relationship between the foci and the vertices.

Step 1: Identify the center

The foci are given as $(-21, 10)$ and $(3, 10)$. Since the center of an ellipse is the midpoint between the foci, we find the center by averaging the coordinates of the foci.

$\text{Center} = \left( \frac{-21 + 3}{2}, \frac{10 + 10}{2} \right) = \left( \frac{-18}{2}, 10 \right) = (-9, 10)$

Thus, the center is $(-9, 10)$.

Step 2: Calculate the distance between the foci

The distance between the two foci is:

$d = \sqrt{(3 - (-21))^2 + (10 - 10)^2} = \sqrt{(3 + 21)^2} = \sqrt{24^2} = 24$

This distance is equal to $2c$, where $c$ is the distance from the center to each focus. So:

$c = \frac{24}{2} = 12$

Step 3: Use the point on the ellipse

The given point $(-9, 15)$ lies on the ellipse. The general equation for an ellipse centered at $(h, k)$ with semi-major axis $a$ and semi-minor axis $b$ is:

$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

From the foci, we know that the major axis is horizontal, meaning the equation will be of the form:

$\frac{(x+9)^2}{a^2} + \frac{(y-10)^2}{b^2} = 1$

Step 4: Find $a^2$

Since $c = 12$, the relationship between $a$, $b$, and $c$ for an ellipse is given by:

$c^2 = a^2 - b^2$

We also need to use the point $(-9, 15)$ to find $a^2$. At this point:

$\frac{(-9+9)^2}{a^2} + \frac{(15-10)^2}{b^2} = 1$

This simplifies to:

$\frac{0}{a^2} + \frac{25}{b^2} = 1$

Thus:

$\frac{25}{b^2} = 1 \implies b^2 = 25$

Step 5: Solve for $a^2$

From the equation $c^2 = a^2 - b^2$, we have:

$12^2 = a^2 - 25 \implies 144 = a^2 - 25 \implies a^2 = 169$

Final Equation

Now that we have $a^2 = 169$ and $b^2 = 25$, the equation of the ellipse is:

$\frac{(x+9)^2}{169} + \frac{(y-10)^2}{25} = 1$

This is the standard equation of the ellipse.

Would you like more details, or do you have any questions? Here are five questions that extend this problem:

1. How do you find the equation if the major axis was vertical?
2. What is the eccentricity of this ellipse?
3. How would the equation change if the ellipse was centered at the origin?
4. Can you find the length of the major and minor axes?
5. What happens to the equation if the foci move closer together?

Tip: The sum of the distances from any point on the ellipse to the two foci is always constant and equals the length of the major axis.