1. **Problem statement:** Find the equation of the ellipse passing through the points $(3,0)$, $(-3,0)$, and $(2,2\sqrt{5})$. 2. **General form of ellipse centered at origin:** $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ where $a$ is the semi-major axis and $b$ is the semi-minor axis. 3. **Use the points on the ellipse:** - For $(3,0)$: $$\frac{3^2}{a^2} + \frac{0^2}{b^2} = 1 \implies \frac{9}{a^2} = 1 \implies a^2 = 9$$ - For $(-3,0)$ (same as above): $$a^2 = 9$$ 4. **Use the point $(2, 2\sqrt{5})$:** $$\frac{2^2}{9} + \frac{(2\sqrt{5})^2}{b^2} = 1$$ $$\frac{4}{9} + \frac{4 \times 5}{b^2} = 1$$ $$\frac{4}{9} + \frac{20}{b^2} = 1$$ 5. **Solve for $b^2$:** $$\frac{20}{b^2} = 1 - \frac{4}{9} = \frac{5}{9}$$ $$b^2 = \frac{20 \times 9}{5} = 36$$ 6. **Final equation of the ellipse:** $$\frac{x^2}{9} + \frac{y^2}{36} = 1$$ This ellipse passes through the given points.