Introduction to Circle & System of Circles for JEE Main

The chapter Circle and System of Circles is a key topic in Class 11 Maths, forming the foundation for coordinate geometry, tangency problems, and conic sections in JEE Main.

A circle is defined as the set of points equidistant from a fixed point (centre). Understanding the equation of circle, tangent, normal, and interaction between multiple circles is essential for solving conceptual and numerical problems efficiently.

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Equation of a Circle

1. Centre-Radius Form:

• Standard form: (x – h)² + (y – k)² = r², where (h, k) = centre, r = radius

2. General Form:

• x² + y² + 2gx + 2fy + c = 0

• Centre = (–g, –f), Radius = √(g² + f² – c)

3. Diameter Form:

• If endpoints of diameter are A(x₁, y₁) and B(x₂, y₂):
  (x – x₁)(x – x₂) + (y – y₁)(y – y₂) = 0

Position of Points and Lines Relative to Circle

1. Point Relative to Circle:

• For circle (x – h)² + (y – k)² = r², point P(x₀, y₀)

  • On circle: (x₀ – h)² + (y₀ – k)² = r²

  • Inside: (x₀ – h)² + (y₀ – k)² < r²

  • Outside: (x₀ – h)² + (y₀ – k)² > r²

2. Line Relative to Circle:

• Distance from centre d = |Ax + By + C| / √(A² + B²)

• Tangent: d = r

• Secant: d < r

• Non-intersecting: d > r

Tangent and Normal to a Circle

1. Tangent Line:

• Line touching circle at exactly one point

• Equation at point P(x₁, y₁) on circle:
  (x – h)(x₁ – h) + (y – k)(y₁ – k) = r²

2. Normal Line:

• Perpendicular to tangent, passes through centre

3. Length of Tangent from External Point (x₀, y₀):

• √[(x₀ – h)² + (y₀ – k)² – r²]

Chord of a Circle

• Segment joining two points on a circle

• Midpoint of chord and perpendicular from centre are key in coordinate geometry

• Equation of chord: T = S₁, where T = xy-term representation, S₁ = value at midpoint

System of Circles

1. Concentric Circles:

• Same centre, different radii

• Equation: (x – h)² + (y – k)² = r₁², r₂²

2. Intersecting Circles:

• Distance between centres d < sum of radii, d > |r₁ – r₂|

• Points of intersection solved using simultaneous equations

3. Tangent Circles:

• Externally tangent: d = r₁ + r₂

• Internally tangent: d = |r₁ – r₂|

4. Coaxial Circles:

• Share common radical axis

• Useful in solving multi-circle intersection and tangency problems

Applications of Circle & System of Circles in JEE Main

• Solve coordinate geometry problems

• Find tangent and normal equations

• Solve intersection points and distance problems

• Solve complex circle system problems in MCQs and numerical

Example: Find tangent to circle x² + y² – 4x – 6y + 9 = 0 at point (2, 1) → equation: (x – 2) + (y – 1) = r²

Important JEE Main Preparation Tips for Circle & System of Circles

• Memorize all forms of circle equations and radius-centre relations

• Practice tangent, normal, chord, and system of circles problems

• Solve simultaneous circle equations for intersection points

• Visualize circle positions and distances for faster solution

• Solve previous year JEE Main MCQs and numerical problems

Why Study Circle & System of Circles from Studentbro.in?

Studentbro.in provides:

• Step-by-step explanations of centre-radius, general, and diameter forms

• Solved examples for tangent, normal, chord, and intersection problems

• Diagrams and charts for quick revision

• MCQs and PYQs aligned with JEE Main syllabus

Conclusion

The Circle & System of Circles chapter is fundamental and scoring in Class 11 Maths for JEE Main. Mastery of centre-radius form, general form, tangent, normal, chord, and circle systems helps students solve conceptual and numerical problems efficiently.

Studentbro.in provides structured, easy-to-understand, and exam-focused content to master Circle & System of Circles effectively and boost JEE Main scores.