# **Exercise Questions** All questions in this exercise are listed below. Click on a question to view its solution. --- --- # **Key Concepts** This exercise focuses on the following concepts: * **Standard and General Equations of Ellipses**: Understanding the algebraic representation of ellipses centered at $(h, k)$. * **Geometric Properties**: Identifying the centre, vertices, co-vertices, and foci. * **Eccentricity**: Calculating the ratio $e = \frac{c}{a}$ which defines the "flatness" of the ellipse. * **Axes Analysis**: Determining the length and equations of the major and minor axes. * **Calculus Applications**: Using differentiation to locate extrema (vertices) on a closed curve. --- # **Important Formulas** Below are the key formulas used in this exercise: | Feature | Horizontal Major Axis ($a > b$) | Vertical Major Axis ($b > a$) | | :--- | :--- | :--- | | **Standard Equation** | $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ | $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ | | **Relationship** | $c^2 = a^2 - b^2$ | $c^2 = b^2 - a^2$ | | **Eccentricity** | $e = \frac{c}{a}$ | $e = \frac{c}{b}$ | | **Foci** | $(h \pm c, k)$ | $(h, k \pm c)$ | | **Vertices** | $(h \pm a, k)$ | $(h, k \pm b)$ | --- # **Summary** This exercise provides a comprehensive review of the ellipse. It transitions from basic identification of properties (centre, foci, eccentricity) to the synthesis of equations from specific data points. A unique aspect of this exercise is the application of differentiation to find vertices, bridging the gap between coordinate geometry and calculus. Key strategies include completing the square to reach standard form and using the relationship $c^2 = a^2 - b^2$ to link the axes lengths to the focal distance.