1. Let's start with the ellipse. An ellipse is the set of all points where the sum of the distances from two fixed points (foci) is constant. 2. The standard form of an ellipse centered at the origin is $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ where $a$ and $b$ are the semi-major and semi-minor axes. 3. Derivation: For any point $(x,y)$ on the ellipse, the sum of distances to foci $(\pm c,0)$ is $2a$, where $c^2 = a^2 - b^2$. 4. Next, the hyperbola is the set of points where the absolute difference of distances from two fixed points (foci) is constant. 5. The standard form of a hyperbola centered at the origin is $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ with $c^2 = a^2 + b^2$. 6. Derivation: For any point $(x,y)$ on the hyperbola, the difference of distances to foci $(\pm c,0)$ is $2a$. 7. The parabola is the set of points equidistant from a fixed point (focus) and a fixed line (directrix). 8. The standard form of a parabola with vertex at origin and focus at $(0,p)$ is $$y^2 = 4px$$ (if horizontal) or $$x^2 = 4py$$ (if vertical). 9. Derivation: Using the distance formula, the distance from $(x,y)$ to focus equals distance to directrix. 10. Diagrams are best visualized graphically; here, the Desmos latex for each curve is provided for plotting. Final formulas: Ellipse: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ Hyperbola: $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ Parabola: $$y^2 = 4px$$ or $$x^2 = 4py$$