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 $S_1>0\Rightarrow$ outside, $S_1=0\Rightarrow$ on, $S_1<0\Rightarrow$ inside |

| Length of tangent from $(x_1,y_1)$ to circle | $L=\sqrt{\Vert S_1\Vert }=\sqrt{\Vert x_1^2+y_1^2+2g{x}_1+2f{y}_1+c\Vert }$ |

| Equation of tangent at point $(x_1,y_1)$ on circle | $x{x}_1+y{y}_1+g(x+x_1)+f(y+y_1)+c=0$ | | Pair of tangents from external point | $S{S}_1=T^2$ where $T=x{x}_1+y{y}_1+g(x+x_1)+f(y+y_1)+c$ |

| Tangential quadrilateral property | $AB+CD=BC+DA$ (sum of opposite sides equal) |

| Tangent-secant theorem | $P{T}^2=PA\cdot PB$ where $PT$ is tangent length and $PAB$ is secant |

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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 $S{S}_1=T^2$ 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.