Circles & System of Circles

Axiomatic Definition

A circle is the locus of a point $P(x,y)$ that moves in a plane such that its distance from a fixed point $C(-g,-f)$ (centre) is a constant radius $r=\sqrt{g^2+f^2-c}$. General Equation: $x^2+y^2+2gx+2fy+c=0$.

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1. Tangents, Normals & Length of Tangent

📊Visual Mathematical Intuition

Circle Tangents, Director Circle & Radical Axis Orthogonality

Director Circle:$x^2 + y^2 = 2r^2 \text{ (Locus of } 90^\circ \text{ Tangent Intersections)}$

Radical Axis:$S_1 - S_2 = 0 \implies \text{Equal Tangent Lengths } L_1 = L_2$

Orthogonal Circles:$2g_1 g_2 + 2f_1 f_2 = c_1 + c_2$

Director circle $x^2 + y^2 = 2r^2$ as the orthogonal locus of perpendicular tangents, and the radical axis $S_1 - S_2 = 0$ perpendicular to the line of centers with equal tangent lengths.

For circle $S\equiv x^2+y^2+2gx+2fy+c=0$ and point $P(x_1,y_1)$:

• Power of Point $P$ / Length of Tangent: $L=\sqrt{S_1}=\sqrt{x_1^2+y_1^2+2g{x}_1+2f{y}_1+c}$
• Equation of Tangent ($T=0$):$$x{x}_1+y{y}_1+g(x+x_1)+f(y+y_1)+c=0$$
• Equation of Normal: Passes through centre $(-g,-f)$ and $(x_1,y_1)$:$$(y_1+f)(x+g)-(x_1+g)(y+f)=0$$
• Chord with Given Midpoint $(x_1,y_1)$: $T=S_1$
• Chord of Contact from External Point $(x_1,y_1)$: $T=0$
• Pair of Tangents from External Point: $S{S}_1=T^2$

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2. Director Circle

Director Circle

The locus of intersection points of two perpendicular tangents to a circle $x^2+y^2=r^2$ is a concentric circle with radius $\sqrt{2}r$:

$$x^2+y^2=2r^2$$

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3. Orthogonality of Two Circles

Orthogonal Intersection Criterion

Two circles $S_1\equiv x^2+y^2+2g_{1}x+2f_{1}y+c_1=0$ and $S_2\equiv x^2+y^2+2g_{2}x+2f_{2}y+c_2=0$ cut each other orthogonally ($\theta ={90}^{\circ }$) if and only if:

$$2g_1{g}_2+2f_1{f}_2=c_1+c_2$$

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4. Radical Axis & Radical Centre

• Radical Axis of $S_1$ and $S_2$: The locus of points having equal lengths of tangents (equal powers) with respect to both circles:$$S_1-S_2=0$$Properties: The radical axis is always perpendicular to the line joining the centres of the two circles.
• Radical Centre: The common intersection point of the radical axes of three mutually non-concentric circles taken in pairs ($S_1-S_2=0,S_2-S_3=0,S_3-S_1=0$).