Circles

Concept overview

A circle with centre (h, k) and radius r has equation (x − h)² + (y − k)² = r². The general form x² + y² + 2gx + 2fy + c = 0 represents a circle with centre (−g, −f) and radius √(g² + f² − c) — real iff g² + f² > c.

A line y = mx + c is tangent to the circle x² + y² = r² iff c² = r²(1 + m²). Equivalently, the perpendicular distance from centre to line equals radius.

Chord of contact: for a point P(x₁, y₁) outside a circle, the pair of tangents from P touches the circle at two points; the chord joining these is the chord of contact, with equation T = 0 where T is obtained by replacing x² → xx₁, y² → yy₁, x → (x + x₁)/2, y → (y + y₁)/2 in the circle equation.

Radical axis of two circles S₁ = 0, S₂ = 0 is S₁ − S₂ = 0 — the locus of points from which tangents to both circles have equal length. It's perpendicular to the line joining the centres.

Family of circles through the intersection of two circles: S₁ + λ S₂ = 0. Family through intersection of a circle and a line: S + λ L = 0.

Key formulas & concepts

• Standard equation: (x − h)² + (y − k)² = r²
• General equation: x² + y² + 2gx + 2fy + c = 0; centre (−g, −f); r = √(g² + f² − c)
• Tangent condition (line vs circle): c² = r²(1 + m²) for x² + y² = r²
• Length of tangent: √S₁ where S₁ = x₁² + y₁² + 2gx₁ + 2fy₁ + c
• Radical axis: S₁ − S₂ = 0
• Family through intersection: S₁ + λ S₂ = 0

Common mistakes to avoid

• Assuming any second-degree equation is a circle — coefficients of x² and y² must be equal and xy term must vanish.
• Ignoring the discriminant condition g² + f² > c for a real circle.
• Confusing chord of contact (from external point) with normal chord.
• Missing that the radical axis is perpendicular to the line of centres.

JEE Main relevance

JEE Main asks 1–2 circle problems every paper — usually one tangent condition and one chord/family problem.

JEE Advanced relevance

Advanced problems combine multiple circles (common chords, orthogonal circles) or use circle theorems inscribed in complex geometry setups.

Previous-year questions (PYQ)

Family of circles and tangent-length problems are the most frequent PYQ archetypes.