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

How do I practise Circle through the intersection of two circles?

1 question with a worked solution

Check before you answer: Parameter Determination: Setting up S + λU = 0 and applying a given third condition (e.g., passing through a point, or having its centre on a specified line) to solve for λ.

1. What is the general Cartesian form representing any circle passing through the points of intersection of a given circle S ≡ x² + y² + 2gx + 2fy + c = 0 and a straight line L ≡ lx + my + n = 0, where λ is a real parameter?
   1. A) S · L = λ
   2. B) S / L = λ
   3. C) S² + λ L² = 0
   4. D) S + λ L = 0
   Check my answer

   Correct answer: D) S + λ L = 0

   The linear combination S + λ L = 0 represents a second-degree curve with identical coefficients for x² and y² and no xy cross-term, which is satisfied by all points where both S = 0 and L = 0 simultaneously. Hence, S + λ L = 0 represents the coaxal family of circles through the intersection points.

   Show worked solution

   Step 1: Any point satisfying both S = 0 and L = 0 satisfies S + λ L = 0. Step 2: S + λ L = 0 retains x² + y² coefficients and no xy term, hence is a circle. Answer: S + λ L = 0.
   Modelled on: the A/L Combined Mathematics examination pattern

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Where does Circle through the intersection of two circles fit in the syllabus?

Year 1 · Term 3