To find the standard equation of an ellipse given the covertices and a focus, let's break down the problem step by step.

Step 1: Identify the center, covertices, and major/minor axes.

• Covertices: (-4, 8) and (10, 8)
  These points lie on the minor axis of the ellipse.
• Focus: (3, 12)

The center of the ellipse lies at the midpoint of the covertices.

Step 2: Calculate the center.

The center $(h, k)$ is the midpoint of the covertices (-4, 8) and (10, 8): $h = \frac{-4 + 10}{2} = \frac{6}{2} = 3$ $k = \frac{8 + 8}{2} = \frac{16}{2} = 8$ So, the center is $(h, k) = (3, 8)$.

Step 3: Determine the lengths of the semi-major and semi-minor axes.

The distance between the covertices gives the length of the minor axis. Since the distance between (-4, 8) and (10, 8) is: $2b = \sqrt{(10 - (-4))^2 + (8 - 8)^2} = \sqrt{(10 + 4)^2} = \sqrt{14^2} = 14$ So, $b = 7$.

Next, use the distance between the center and the focus to find $c$: $c = \sqrt{(3 - 3)^2 + (12 - 8)^2} = \sqrt{0 + 4^2} = 4$

Finally, use the relationship $c^2 = a^2 - b^2$ to find $a$, the length of the semi-major axis: $c^2 = a^2 - b^2$ $4^2 = a^2 - 7^2$ $16 = a^2 - 49$ $a^2 = 65$ So, $a = \sqrt{65}$.

Step 4: Write the standard equation of the ellipse.

Since the major axis is vertical (the foci and the center share the same x-coordinate), the standard form of the ellipse's equation is: $\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1$ Substituting $h = 3$, $k = 8$, $a^2 = 65$, and $b^2 = 49$: $\frac{(x - 3)^2}{49} + \frac{(y - 8)^2}{65} = 1$

So, the standard equation of the ellipse is: $\boxed{\frac{(x - 3)^2}{49} + \frac{(y - 8)^2}{65} = 1}$

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:

1. How do you find the equation of an ellipse given its vertices and a point on the ellipse?
2. What is the relationship between the foci and the major/minor axes in an ellipse?
3. How do you determine the length of the major axis if only the foci and covertices are known?
4. How would the equation of the ellipse change if the major axis was horizontal instead of vertical?
5. How do you find the eccentricity of an ellipse, and what does it represent?

Tip: The value of $a^2$ in an ellipse is always associated with the larger of the two denominators, which corresponds to the major axis.