We are given the following information for an ellipse:

• Co-vertices: $(4, 8)$ and $(10, 8)$
• One focus: $(3, 12)$
• Center: $(7, 8)$

Step 1: Determine the center of the ellipse

The center of the ellipse is directly given as $(7, 8)$.

Step 2: Determine the distance from the center to the co-vertices

The co-vertices are at $(4, 8)$ and $(10, 8)$. These lie along the minor axis of the ellipse. The minor axis is horizontal because both points have the same $y$-coordinate, so the distance between the co-vertices is the length of the minor axis.

The distance between the center and a co-vertex is the semi-minor axis $b$: $b = \frac{10 - 4}{2} = 3$ Thus, the length of the semi-minor axis is $b = 3$.

Step 3: Determine the distance from the center to the focus

The focus lies at $(3, 12)$. The distance between the center $(7, 8)$ and the focus is the distance along the major axis, which we can calculate using the distance formula: $c = \sqrt{(7 - 3)^2 + (8 - 12)^2} = \sqrt{(4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}$ Thus, $c = 4\sqrt{2}$.

Step 4: Use the relationship between $a$, $b$, and $c$

In an ellipse, the relationship between the semi-major axis $a$, semi-minor axis $b$, and the focal distance $c$ is given by: $c^2 = a^2 - b^2$ Substitute $c = 4\sqrt{2}$ and $b = 3$ into this equation: $(4\sqrt{2})^2 = a^2 - 3^2$ $32 = a^2 - 9$ $a^2 = 41$ Thus, $a = \sqrt{41}$.

Step 5: Write the standard form of the equation

The standard form of the equation of an ellipse is: $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$ where $(h, k)$ is the center, $a$ is the length of the semi-major axis, and $b$ is the length of the semi-minor axis.

Substitute $h = 7$, $k = 8$, $a^2 = 41$, and $b^2 = 9$: $\frac{(x - 7)^2}{41} + \frac{(y - 8)^2}{9} = 1$

Thus, the equation of the ellipse in standard form is: $\boxed{\frac{(x - 7)^2}{41} + \frac{(y - 8)^2}{9} = 1}$

Would you like more details or have any other questions?

Here are 5 related questions to expand your understanding:

1. How do you find the foci of an ellipse if the equation is given?
2. What is the geometric interpretation of $a$, $b$, and $c$ in an ellipse?
3. How do you identify the major and minor axes from an ellipse's equation?
4. Can an ellipse have its major axis along the $y$-axis instead of the $x$-axis?
5. What are the key differences between an ellipse and a hyperbola?

Tip: Always verify the center and orientation of an ellipse from its co-vertices and foci before writing the equation.