The problem describes the set $\mathcal{E}$ as the locus of points $P = (x, y)$ in the Cartesian plane such that the sum of the distances to two fixed points $F_1 = (0, 4)$ and $F_2 = (0, -4)$ is equal to 10.

This setup is the definition of an ellipse. For an ellipse, the sum of the distances from any point on the ellipse to the two foci is constant and equal to the length of the major axis. Let’s find the equation of this ellipse.

Step-by-Step Solution

1. Identify the foci and the sum of distances:

   • The foci are $F_1 = (0, 4)$ and $F_2 = (0, -4)$.
   • The sum of distances is 10.

2. Use the standard definition of an ellipse: The equation of an ellipse is given by: $d(P, F_1) + d(P, F_2) = 2a$ where $d(P, F_1)$ and $d(P, F_2)$ are the distances from point $P = (x, y)$ to the foci $F_1$ and $F_2$, and $2a$ is the length of the major axis. Here, $2a = 10$, so $a = 5$.

3. Distance formulas:

   • $d(P, F_1) = \sqrt{x^2 + (y - 4)^2}$
   • $d(P, F_2) = \sqrt{x^2 + (y + 4)^2}$

4. Set up the equation: $\sqrt{x^2 + (y - 4)^2} + \sqrt{x^2 + (y + 4)^2} = 10$

This is the equation representing the locus of points on the ellipse.

If you want, we can further simplify or manipulate this equation into a standard form, but this already represents the equation of the ellipse.

Would you like more details or have any further questions?

Here are 5 related questions to expand your understanding:

1. What is the general equation of an ellipse centered at the origin with a vertical major axis?
2. How do you derive the equation of an ellipse given its foci and the length of the major axis?
3. What would change if the sum of distances was less than 10? Would it still form an ellipse?
4. How do you find the lengths of the semi-major and semi-minor axes of an ellipse?
5. How can you graph an ellipse given its foci and the sum of distances?

Tip: Remember, the definition of an ellipse is based on the sum of the distances from any point on the curve to the two foci being constant.