1. **State the problem:** Find the equation of an ellipse given the center $(1,3)$, one focus $(1,0)$, and one vertex $(1,-1)$. 2. **Recall the standard form of an ellipse equation:** If the ellipse is vertical (major axis along $y$-axis), the equation is $$\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$$ where $(h,k)$ is the center, $a$ is the semi-major axis length, and $b$ is the semi-minor axis length. 3. **Identify given values:** - Center: $(h,k) = (1,3)$ - One vertex: $(1,-1)$ lies on the major axis, so $a = |3 - (-1)| = 4$ - One focus: $(1,0)$ 4. **Calculate $c$, the distance from center to focus:** $$c = |3 - 0| = 3$$ 5. **Use the relationship between $a$, $b$, and $c$ for ellipses:** $$c^2 = a^2 - b^2$$ Substitute known values: $$3^2 = 4^2 - b^2$$ $$9 = 16 - b^2$$ 6. **Solve for $b^2$:** $$b^2 = 16 - 9 = 7$$ 7. **Write the equation of the ellipse:** $$\frac{(x-1)^2}{7} + \frac{(y-3)^2}{16} = 1$$ **Final answer:** The equation of the ellipse is $$\frac{(x-1)^2}{7} + \frac{(y-3)^2}{16} = 1$$