# **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: - Intersection of lines and circles - Tangents to circles (equations and conditions) - Normals to circles - Condition for tangency: distance from center equals radius - Perpendicularity of tangent and radius - Chords and diameters of circles - Condition for two circles to touch (internally or externally) --- # **Important Formulas** Below are the key formulas used in this exercise: **Standard Circle Equation:** $$(x-h)^2 + (y-k)^2 = r^2$$ **General Circle Equation:** $$x^2 + y^2 + 2gx + 2fy + c = 0$$ with center $(-g, -f)$ and radius $\sqrt{g^2 + f^2 - c}$ **Condition for Line to be Tangent to Circle:** $$d = r$$ where $d$ is perpendicular distance from center to line **Equation of Tangent to Circle $x^2 + y^2 = a^2$ at $(x_1, y_1)$:** $$xx_1 + yy_1 = a^2$$ **Equation of Normal to Circle at $(x_1, y_1)$:** $$y - y_1 = \frac{y_1}{x_1}(x - x_1)$$ **Condition for Two Circles to Touch:** $$|C_1C_2| = r_1 \pm r_2$$ (external: $+$ , internal: $-$) --- # **Summary** This exercise covers fundamental properties of circles and their interactions with lines. Key strategies include: converting general form to standard form to identify center and radius; using the distance formula to establish tangency conditions; applying the geometric property that tangent is perpendicular to radius; and using the fact that the normal passes through the center. For intersection problems, simultaneous equations are essential. When circles touch, comparing the distance between centers with sum/difference of radii determines the nature of contact.