To find the correct equation of the ellipse from the provided graph, we need to carefully examine the key details:

1. Orientation: The ellipse is vertically oriented (as the vertices are aligned along the y-axis).
2. Center: The center of the ellipse appears to be $(6, 7)$ based on the symmetry of the points.
3. Vertices: The major axis extends vertically, with one vertex around $(6, 17)$ and the other near $(6, -3)$, indicating the length of the semi-major axis is 10 units.
4. Foci: The foci seem to be located on the same vertical line (y-axis), slightly closer to the center than the vertices.

Ellipse Standard Form

For an ellipse with a vertical major axis, the equation is:

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

• $(h, k)$ is the center,
• $a$ is the length of the semi-major axis,
• $b$ is the length of the semi-minor axis.

From the Graph:

• The center is $(6, 7)$,
• The distance from the center to the vertex (semi-major axis) is $a = 10$,
• The semi-minor axis length $b$ is found from one of the smaller axes. It seems to be 12 units.

Thus, the equation is:

$\frac{(x - 6)^2}{144} + \frac{(y - 7)^2}{169} = 1$

This matches the fourth option on the list.

Final Answer:

The correct equation is $\frac{(x - 6)^2}{144} + \frac{(y - 7)^2}{169} = 1$.

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. How is the equation of an ellipse derived from its geometric properties?
2. What are the differences between ellipses with horizontal and vertical major axes?
3. How do you find the foci of an ellipse?
4. What is the relationship between the lengths of the semi-major axis, semi-minor axis, and the distance between the foci?
5. How would the equation change if the center of the ellipse shifted?

Tip: Remember, the distance between the foci and center is $\sqrt{a^2 - b^2}$, where $a$ is the semi-major axis and $b$ is the semi-minor axis.