Circle Through Points of Intersection (JEE Advanced)

“equation of a circle through the points of intersection of two circles and those of a circle and a straight line”

Source: JEE Advanced 2026 Information Bulletin, Annexure-I. Closest JEE Main topic: points of intersection of a line and a circle with the centre at the origin.

JEE Main only covers line-circle intersections with circles centered at the origin, which is a simpler special case. For Advanced, you need to handle general circles and the powerful S₁ + λS₂ = 0 family approach, which Main students have never practiced.

• If S₁ = 0 and S₂ = 0 are two circles, then S₁ + λS₂ = 0 represents a family of circles through their intersection points (for all λ ≠ -1)
• If S = 0 is a circle and L = 0 is a line, then S + λL = 0 represents a family of circles through the points where the line cuts the circle
• The radical axis of two circles S₁ = 0 and S₂ = 0 is given by S₁ - S₂ = 0, which is always a straight line
• To find a specific circle from the family, use an extra condition like passing through a given point, having a specific radius, or having center on a given line
• When λ = -1 in S₁ + λS₂ = 0, you get the radical axis (a line), not a circle

• S₁ + λS₂ = 0 : family of circles through intersection of circles S₁ = 0 and S₂ = 0
• S + λL = 0 : family of circles through intersection of circle S = 0 and line L = 0
• Radical axis: S₁ - S₂ = 0 (set λ = -1)
• General circle: x² + y² + 2gx + 2fy + c = 0 with center (-g, -f) and radius √(g² + f² - c)
• For S₁ + λS₂ = 0 to be a circle, coefficient of x² must equal coefficient of y² and xy term must be zero
• Two circles intersect if |r₁ - r₂| < d < r₁ + r₂ where d is distance between centers

• Forgetting that λ = -1 gives a line (radical axis), not a circle, so this value must be excluded when finding circle families
• Not normalizing the circle equations before applying the formula; both S₁ and S₂ should have coefficient 1 for x² and y²
• Confusing S₁ + λS₂ = 0 with S₁ - S₂ = 0; the first is a family of circles, the second is specifically the radical axis
• Assuming the intersection points must be found explicitly; the whole point of this method is to avoid that calculation

Question. Find the equation of the circle passing through the points of intersection of the circles x² + y² - 4x - 6y + 9 = 0 and x² + y² - 2x - 4y + 1 = 0, and also passing through the origin.

Solved twice independently; both solutions agree.