# **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: - Position of a point relative to a circle (inside, outside, or on) - Length of tangent from an external point to a circle - Equation of tangent lines from external points - Properties of tangential quadrilaterals (Pitot's theorem) - Power of a point (tangent-secant theorem) - Algebraic conditions for tangency ($S_1$ criterion) --- # **Important Formulas** Below are the key formulas used in this exercise: | Concept | Formula | | :--- | :--- | | **Position of point** $(x_1, y_1)$ relative to circle $S=0$ | $S_1 = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c$
$S_1 > 0 \Rightarrow$ outside, $S_1 = 0 \Rightarrow$ on, $S_1 < 0 \Rightarrow$ inside | | **Length of tangent** from $(x_1, y_1)$ to circle | $L = \sqrt{\|S_1\|} = \sqrt{\|x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c\|}$ | | **Equation of tangent** at point $(x_1, y_1)$ on circle | $xx_1 + yy_1 + g(x+x_1) + f(y+y_1) + c = 0$ | | **Pair of tangents** from external point | $SS_1 = T^2$ where $T = xx_1 + yy_1 + g(x+x_1) + f(y+y_1) + c$ | | **Tangential quadrilateral** property | $AB + CD = BC + DA$ (sum of opposite sides equal) | | **Tangent-secant theorem** | $PT^2 = PA \cdot PB$ where $PT$ is tangent length and $PAB$ is secant | --- # **Summary** This exercise covers fundamental properties of circles and tangents. Key skills include: determining point positions using the $S_1$ algebraic criterion; calculating tangent lengths via the power-of-a-point formula $L = \sqrt{S_1}$; finding equations of tangents from external points using the $SS_1 = T^2$ condition; proving geometric properties of quadrilaterals circumscribing circles (Pitot's theorem); and applying the tangent-secant theorem relating tangent and secant segments from an external point. Common strategies involve converting general circle equations to standard form $(x-h)^2 + (y-k)^2 = r^2$ and recognizing when to apply geometric theorems versus algebraic coordinate geometry.