To find the standard form equation of the ellipse, let's determine the type of ellipse, its center, and the relationship between its components.

Step 1: Identify key elements

1. Vertices: (0, ±7). This tells us the major axis is vertical, centered at (0, 0), and the semi-major axis $a = 7$.
2. Foci: (0, ±$\sqrt{15}$). This confirms that the foci are also on the vertical axis, so the ellipse's equation is of the form: $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$
3. Relationship between $a$, $b$, and $c$: For an ellipse: $c^2 = a^2 - b^2$ where:
   • $c$ is the distance from the center to each focus,
   • $a$ is the semi-major axis,
   • $b$ is the semi-minor axis.

Step 2: Solve for $b^2$

Given:

• $a = 7$, so $a^2 = 49$,
• $c = \sqrt{15}$, so $c^2 = 15$.

Using the relationship $c^2 = a^2 - b^2$: $15 = 49 - b^2$ $b^2 = 49 - 15 = 34$

Step 3: Write the equation

Substitute $a^2 = 49$ and $b^2 = 34$ into the standard form: $\frac{x^2}{34} + \frac{y^2}{49} = 1$

Final Answer:

$\frac{x^2}{34} + \frac{y^2}{49} = 1$

Would you like a detailed explanation of any step or further clarification?

5 Related Questions:

1. How do you determine whether an ellipse is vertical or horizontal?
2. What is the relationship between the foci and the semi-major axis of an ellipse?
3. How do you find the foci of an ellipse from its standard form equation?
4. What distinguishes an ellipse from a hyperbola in terms of their equations?
5. Can you derive the equation of an ellipse if given its eccentricity?

Tip:

Always remember that $a^2$ is associated with the larger denominator in the standard form equation of an ellipse, regardless of whether it's horizontal or vertical.