Circles

🟢 Lite — Quick Review

Rapid summary for last-minute revision before your exam.

Circles — Key Facts for JEE Main Standard form: (x − h)² + (y − k)² = r² with centre (h, k), radius r General form: x² + y² + 2gx + 2fy + c = 0 with centre (−g, −f), radius = √(g² + f² − c) Equation of tangent at point (x₁, y₁) on circle: use T = 0 formula Length of tangent from external point P: √[SP² − r²] where S is circle equation value ⚡ Exam tip: JEE Main loves circle tangent problems — remember T = 0 for tangent at point on circle and S₁ = 0 for condition of point lying on circle!

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🟡 Standard — Core Study

Standard content for students with a few days to months.

Circles — JEE Main Study Guide

Standard forms:

• Centre (h, k), radius r: (x − h)² + (y − k)² = r²
• Centre at origin: x² + y² = r²
• General form: x² + y² + 2gx + 2fy + c = 0 → centre (−g, −f), radius √(g² + f² − c)
• Condition for real circle: g² + f² > c

Parametric form: x = h + r cos θ; y = k + r sin θ; point on circle is (h + r cos θ, k + r sin θ)

Tangent at point (x₁, y₁) on circle: For circle S ≡ x² + y² + 2gx + 2fy + c = 0: use T ≡ xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0 For (x − h)² + (y − k)² = r²: use T ≡ (x − h)(x₁ − h) + (y − k)(y₁ − k) = r²

Condition of tangency: Line y = mx + c touches circle x² + y² = r² if c² = r²(1 + m²) Line y = mx + c touches circle (x − h)² + (y − k)² = r² if (c − k)² = r²(1 + m²) − (mh − k)²… actually use distance formula from centre to line = r

Length of tangent: From external point (x₁, y₁) to circle S = 0: length = √[S₁₁] where S₁₁ = x₁² + y₁² + 2gx₁ + 2fy₁ + c = √[(distance from point to centre)² − r²]

Point inside/outside circle: If S₁₁ < 0: point lies inside; S₁₁ > 0: outside; S₁₁ = 0: on circle

Equation of circle from endpoints of diameter: If endpoints are (x₁, y₁) and (x₂, y₂): (x − x₁)(x − x₂) + (y − y₁)(y − y₂) = 0

Family of circles:

• Through intersection of two circles S₁ = 0 and S₂ = 0: S₁ + λS₂ = 0
• Through two points: circle with diameter as given points + any circle through that diameter
• Orthogonal circles: S₁ + S₂ = 0 represents circles that cut each other at right angles

Number of circles through given points:

• 3 non-collinear points → one unique circle

• 2 points → infinite circles (family of circles with those points as diameter endpoints)

• 1 point → infinite circles

• Key formula: Centre (−g, −f), radius √(g²+f²−c) for x²+y²+2gx+2fy+c=0; T = 0 for tangent

• Common trap: In general form, radius = √(g² + f² − c), not just √(g² + f²)

• Exam weight: 1–2 questions per year (4–8 marks); frequently combined with straight lines

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🔴 Extended — Deep Dive

Comprehensive coverage for students on a longer study timeline.

Circles — Comprehensive JEE Main Notes

Director circle: Locus of point from which two perpendicular tangents can be drawn to circle x² + y² = r² Equation: x² + y² = 2r²; all points on this circle have tangents that are perpendicular

Chord of contact: From point (x₁, y₁) outside circle, chord of contact (tangent from that point touching circle) is given by T = 0

Polar and pole: For circle S = 0, polar of point P(x₁, y₁) is line T = 0 If point lies on polar of another point, then that point lies on polar of the first (reciprocal property)

Equation of common chord of two circles: S₁ = 0 and S₂ = 0 → common chord: S₁ − S₂ = 0 (a straight line)

Radical axis: Line joining points of equal power with respect to two circles S₁ − S₂ = 0 is the radical axis For three circles, radical axes are concurrent at radical centre

Radical centre: Intersection of radical axes of three circles; common point with equal power to all three

Common tangents: For two circles with centres C₁, C₂ and radii r₁, r₂:

• Direct common tangents: don’t cross the line joining centres
• Transverse common tangents: cross the line joining centres
• Number of common tangents depends on distance d = C₁C₂:
  • d > r₁ + r₂: 4 tangents (2 direct, 2 transverse)
  • d = r₁ + r₂: 3 tangents (2 direct, 1 transverse — touch externally)
  • |r₁ − r₂| < d < r₁ + r₂: 2 tangents (direct only)
  • d = |r₁ − r₂|: 1 tangent (transverse only — touch internally)
  • d < |r₁ − r₂|: 0 common tangents (one inside other)

Image of circle in line: If circle has centre (h, k) and radius r, image in line Ax + By + C = 0 has centre at image point and same radius

Smallest circle containing points: For 3 points: circumcircle (perpendicular bisectors) For 2 points: circle with that diameter

Angle between tangent and chord equals angle in alternate segment: If chord AB and tangent at A: angle between tangent and chord = angle in opposite arc

Parametric forms:

• On x² + y² = r²: (r cos θ, r sin θ)
• On (x − h)² + (y − k)² = r²: (h + r cos θ, k + r sin θ)

Equation of chord with given midpoint: For circle S = 0, chord with midpoint (x₁, y₁): T = S₁ i.e., xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = (x₁² + y₁² + 2gx₁ + 2fy₁ + c)

Intersection of line and circle: Substitute line into circle, solve quadratic in x; if discriminant = 0 → tangent; > 0 → secant; < 0 → no intersection

• Remember: Centre (−g, −f), radius √(g²+f²−c) for general form; T = 0 gives tangent; S₁ = 0 checks if point lies on circle; chord of contact uses T = 0 from external point; common chord of S₁ = 0, S₂ = 0 is S₁ − S₂ = 0
• Previous years: “Find equation of circle through (1,2) and (3,4) with centre on x-axis” [2023]; “Find length of tangent from (5,4) to x²+y²−6x−8y+5=0” [2024]; “Find equation of common chord of x²+y²=9 and x²+y²−6x−8y+5=0” [2024]

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📊 JEE Main Exam Essentials

Detail	Value
Questions	90 (30 per subject)
Time	3 hours
Marks	300 (90 per subject)
Section	Physics (30), Chemistry (30), Mathematics (30)
Negative	−1 for wrong answer
Mode	Computer-based

🎯 High-Yield Topics for JEE Main Mathematics

• Calculus (Differentiation + Integration) — ~35 marks combined
• Coordinate Geometry (straight lines, circles, conics) — ~20 marks
• Algebra (Complex Numbers, Quadratics, P&C, Probability) — ~25 marks
• Trigonometry + Inverse Trigonometry — ~15 marks
• Vector + 3D — ~15 marks

📝 Previous Year Question Patterns

• Circles: 1–2 questions per year, 4–8 marks
• Common patterns: find circle equation, tangent at point, length of tangent, common tangents
• Weight: medium-high frequency, highly scoring

💡 Pro Tips

• For circle problems, always first find centre and radius
• Remember T = 0 for tangent at point on circle and S₁ = 0 for point on circle condition
• For chord of contact from (x₁, y₁): use T = 0 with (x₁, y₁) substituted
• When checking number of common tangents, always compute d = distance between centres first
• Radical axis S₁ − S₂ = 0 is very useful for finding common chord of two circles
• For circle through two points, the centre always lies on perpendicular bisector of the segment

🔗 Official Resources

• NTA Official JEE Main
• JEE Main Syllabus PDF

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