This problem is asking for the equation of a geometric locus $\mathcal{E}$, where the sum of the distances from a point $P = (x, y)$ to two fixed points $F_1 = (0, 4)$ and $F_2 = (0, -4)$ is equal to 10.

This setup is a classic description of an ellipse, where the sum of the distances from any point on the ellipse to the two foci is constant.

Step-by-Step Solution:

1. Definition of an Ellipse: The general equation for an ellipse with two foci is: $d(P, F_1) + d(P, F_2) = 2a$ where $d(P, F_1)$ and $d(P, F_2)$ are the distances from the point $P$ to the foci $F_1$ and $F_2$, and $2a$ is the constant sum of these distances. In this case, $2a = 10$, so $a = 5$.

2. Distance Formula: The distance from the point $P = (x, y)$ to the foci $F_1 = (0, 4)$ and $F_2 = (0, -4)$ can be expressed as: $d(P, F_1) = \sqrt{x^2 + (y - 4)^2}$ $d(P, F_2) = \sqrt{x^2 + (y + 4)^2}$

3. Equation of the Ellipse: The sum of the distances from $P$ to $F_1$ and $F_2$ is given as 10: $\sqrt{x^2 + (y - 4)^2} + \sqrt{x^2 + (y + 4)^2} = 10$

4. Simplifying: To find the explicit equation of the ellipse, we need to square both sides to eliminate the square roots. However, this will result in a more complex algebraic process. Let’s square both sides carefully: $\left( \sqrt{x^2 + (y - 4)^2} + \sqrt{x^2 + (y + 4)^2} \right)^2 = 10^2$ Expanding the left side: $x^2 + (y - 4)^2 + x^2 + (y + 4)^2 + 2\sqrt{(x^2 + (y - 4)^2)(x^2 + (y + 4)^2)} = 100$

   After expanding and simplifying, you should end up with the equation of the ellipse in standard form: $\frac{x^2}{25} + \frac{y^2}{9} = 1$ This is the equation of the ellipse centered at the origin, with semi-major axis $a = 5$ and semi-minor axis $b = 3$.

Final Answer:

The equation of the geometric locus $\mathcal{E}$ is: $\frac{x^2}{25} + \frac{y^2}{9} = 1$

Would you like a further breakdown of the algebra involved in solving this? Here are five related questions to deepen understanding:

1. What is the general form of the equation of an ellipse with foci at specific points?
2. How do we derive the equation of an ellipse from the distance definition?
3. Why is it necessary to square both sides when simplifying the equation?
4. What is the geometric interpretation of the foci and the sum of distances being constant?
5. How would the equation change if the sum of distances was not 10 but another value?

Tip: When working with ellipses, the foci are always located along the major axis, and the constant sum of distances helps determine the shape of the ellipse.