Circle through the intersection of two circles — A/L Combined Mathematics

What must I remember about Circle through the intersection of two circles?

• Pencil / Family of Circles through intersection of two circles: Let S1 = 0 and S2 = 0 be two non-concentric intersecting circles; the equation S1 + λ*S2 = 0 represents a family of circles passing through the common intersection points for all real λ ≠ -1.
• Common Chord (Radical Axis): When λ = -1, second-degree quadratic terms x² and y² cancel, yielding the linear equation S1 - S2 = 0: 2(g1 - g2)x + 2(f1 - f2)y + (c1 - c2) = 0.
• Radical Axis fundamental property: The locus of a point from which the lengths of tangents drawn to two circles S1 = 0 and S2 = 0 are equal (L1 = L2 <=> S1 = S2 <=> S1 - S2 = 0); perpendicular to line connecting the two centers C1C2.
• Circle passing through intersection of a circle and a line: Let circle be S = 0 and line be L = 0; then S + λ*L = 0 represents a family of circles passing through the intersection points of the circle and line for any real λ.
• Radical Center: The radical axes of three circles taken in pairs intersect at a single common point called the Radical Center, from which equal-length tangents can be drawn to all three circles.

Test yourself (1 minute)

1. Determining λ from auxiliary condition

   Substitute a given external point, or use condition of passing through origin, or orthogonality with a third circle to determine parameter λ.

2. Smallest circle through intersection of circle and line

   The circle in the family S + λ*L = 0 having minimum radius is the one where the intercepted chord acts as its DIAMETER (center lies on line L = 0).

3. Radical axis exists even for non-intersecting circles

   S1 - S2 = 0 is the geometric radical axis of equal tangent lengths regardless of whether the circles intersect, touch, or are completely disjoint.

Practise Circle through the intersection of two circles questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 19

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Year 1 · Term 3