To find the standard form of the ellipse given the foci and the sum of the distances from any point on the ellipse to the foci, follow these steps:

Key Information:

• Foci: $(-2, 0)$ and $(4, 0)$
• Sum of distances from any point on the ellipse to the foci = $8$

Step 1: Determine the center

The center $(h, k)$ of the ellipse is the midpoint of the line segment connecting the foci: $h = \frac{-2 + 4}{2} = 1, \quad k = \frac{0 + 0}{2} = 0$ So, the center is at $(1, 0)$.

Step 2: Determine the distance between the foci and semi-major axis

• Distance between the foci (2c): $2c = \sqrt{(4 - (-2))^2 + (0 - 0)^2} = \sqrt{6^2} = 6$ $c = 3$

• Sum of distances is equal to the length of the major axis $2a$: $2a = 8 \quad \Rightarrow \quad a = 4$

Step 3: Find the semi-minor axis

• Use the relationship $c^2 = a^2 - b^2$ to find $b$: $3^2 = 4^2 - b^2 \implies 9 = 16 - b^2 \implies b^2 = 7$

Step 4: Write the standard form equation

Since the major axis is horizontal, the standard form of the ellipse is: $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$ Substituting the values: $\frac{(x - 1)^2}{16} + \frac{y^2}{7} = 1$

So, the standard form of the ellipse is: $\boxed{\frac{(x - 1)^2}{16} + \frac{y^2}{7} = 1}$

Would you like more details or have any questions?

Here are five related questions for further exploration:

1. How would the equation of the ellipse change if the foci were vertically aligned?
2. What are the lengths of the major and minor axes of this ellipse?
3. How can you graph this ellipse using its standard form?
4. How would you find the eccentricity of this ellipse?
5. What is the relationship between the sum of distances and the foci in the context of an ellipse?

Tip: For any ellipse, the sum of the distances from any point on the ellipse to the two foci is always equal to the length of the major axis.