To write the standard form of the ellipse, we need to identify key information about the ellipse from the given dataTo write the standard form of the ellipse, we need to identify key information about the ellipse from the given

Key Information:

• Foci: $(0, -7)$ and $(0, 3)$
• Vertices: $(0, 5)$ and $(0, -9)$

Step 1: Determine the center

The center $(h, k)$ of the ellipse is the midpoint of the line segment connecting the vertices: $k = \frac{5 + (-9)}{2} = \frac{-4}{2} = -2$ Since the x-coordinates of the vertices and foci are both 0, the center is at $(0, -2)$.

Step 2: Determine the lengths of the semi-major and semi-minor axes

• Semi-major axis (a): The distance from the center to one of the vertices. $a = 5 - (-2) = 7$
• Distance between the foci (2c): The distance from the center to one of the foci. $c = 3 - (-2) = 5$
• Relationship between a, b, and c: The equation $c^2 = a^2 - b^2$. $5^2 = 7^2 - b^2 \implies 25 = 49 - b^2 \implies b^2 = 24$

Step 3: Write the standard form equation

Since the ellipse is vertical (the vertices and foci align vertically), the standard form of the ellipse is: $\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1$ Substituting the values we found: $\frac{(x - 0)^2}{24} + \frac{(y + 2)^2}{49} = 1$

So, the standard form of the ellipse is: $\boxed{\frac{x^2}{24} + \frac{(y + 2)^2}{49} = 1}$

Would you like more details or have any questions?

Here are five related questions for further exploration:

1. How would the equation change if the ellipse were horizontal instead of vertical?
2. What are the equations of the major and minor axes of this ellipse?
3. How do you find the length of the latus rectum for this ellipse?
4. Can you derive the formula for the area of this ellipse?
5. How does the equation of a hyperbola differ from that of an ellipse?

Tip: In ellipses, always remember that $a > b$ where $a$ is the semi-major axis and $b$ is the semi-minor axis.