1. **State the problem:** We are given the equation $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ which represents an ellipse centered at the origin. 2. **Formula and explanation:** This is the standard form of an ellipse equation with semi-major axis $a$ and semi-minor axis $b$. The ellipse touches the rectangle formed by $x=\pm a$ and $y=\pm b$. 3. **Important rules:** - The ellipse is symmetric about both axes. - The points where the ellipse touches the rectangle are $(\pm a,0)$ and $(0,\pm b)$. 4. **Intermediate work:** No further simplification is needed as the equation is already in standard form. 5. **Interpretation:** The ellipse is centered at the origin, with horizontal radius $a$ and vertical radius $b$. The rectangle described is the bounding box of the ellipse. **Final answer:** The equation $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ describes an ellipse centered at the origin with semi-major axis $a$ and semi-minor axis $b$.