# **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: - Conic Sections (Circle, Parabola, Ellipse, Hyperbola) - Standard equations and parametric forms of conics - Geometric properties: foci, directrices, vertices, and eccentricity - Real-world applications of conic sections in physics and engineering - Locus problems and coordinate geometry - Circumference and rotational motion calculations --- # **Important Formulas** Below are the key formulas used in this exercise: **Circle:** $$(x-h)^2 + (y-k)^2 = r^2$$ **Parabola (standard forms):** $$y^2 = 4ax \quad \text{or} \quad x^2 = 4ay$$ **Ellipse:** $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad \text{where} \quad c^2 = a^2 - b^2$$ $$\text{Eccentricity: } e = \frac{c}{a} < 1$$ **Hyperbola:** $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \quad \text{where} \quad c^2 = a^2 + b^2$$ $$\text{Eccentricity: } e = \frac{c}{a} > 1$$ **Circumference and Linear Speed:** $$C = 2\pi r$$ $$\text{Speed} = 2\pi r \times \text{revolutions per unit time}$$ --- # **Summary** This exercise demonstrates practical applications of conic sections across physics, engineering, and architecture. Key scenarios include using parabolic trajectories for projectile motion (cricket ball), elliptical properties for planetary orbits and acoustic phenomena (whispering galleries), hyperbolic navigation systems (control towers/LORAN), and circular motion for rotational speed calculations. The problems require translating physical constraints into standard conic equations, utilizing eccentricity to relate parameters, and applying geometric properties (foci, directrices) to solve for unknown distances and positions.