1. **Problem Statement:** Given two points (foci) of an ellipse, find the equation of the ellipse. 2. **Ellipse Definition:** An ellipse is the set of points where the sum of distances to the two foci is constant. 3. **Formula:** If the foci are at $(-c,0)$ and $(c,0)$ on the x-axis, the ellipse equation is $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ where $a > c$ and $b^2 = a^2 - c^2$. 4. **Step-by-step:** - Identify the foci coordinates $(-c,0)$ and $(c,0)$. - Calculate $c$ as the distance from the center to a focus. - Determine the constant sum of distances $2a$ (major axis length). - Compute $b^2 = a^2 - c^2$. 5. **Example:** Suppose foci at $(-3,0)$ and $(3,0)$ and sum of distances $2a = 10$. - Then $c=3$, $a=5$. - Calculate $b^2 = 5^2 - 3^2 = 25 - 9 = 16$. 6. **Final equation:** $$\frac{x^2}{25} + \frac{y^2}{16} = 1$$ This is the ellipse equation with given foci and sum of distances.