# **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 a circle - Centre and radius determination - Circle passing through points (3-point form, 2-point with constraints) - Tangency conditions (distance from centre to line equals radius) - Circles touching coordinate axes - Diameters of a circle and their equations - Family of circles through intersection of two circles - Geometric constraints (lines, rectangles, areas) --- # **Important Formulas** Below are the key formulas used in this exercise: **Standard Form:** $$(x-h)^2 + (y-k)^2 = r^2$$ where $(h,k)$ is the centre and $r$ is the radius. **General Form:** $$x^2 + y^2 + 2gx + 2fy + c = 0$$ - Centre: $(-g, -f)$ - Radius: $\sqrt{g^2 + f^2 - c}$ **Distance from Point to Line:** $$d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}$$ **Condition for Tangency:** $$d = r$$ (Distance from centre to tangent line equals radius) --- # **Summary** This exercise comprehensively covers methods for determining the equation of a circle under various geometric constraints. Key strategies include using the standard form when centre and radius are known or can be derived, converting between general and standard forms, and applying the tangency condition (distance from centre to line equals radius) for problems involving tangent lines. Common patterns involve solving systems of equations to find unknown parameters (centre coordinates or radius) using given points, tangent conditions, or diameter equations. Special cases include circles touching both axes (where $|h|=|k|=r$) and finding equations through the intersection of existing circles using the family of circles approach.