JEE/Mathematics/Circle

Coordinate Geometry · High Yield

Circle — JEE Main & Advanced Notes

Equation, tangent, normal, chord of contact and family of circles are frequent JEE coordinate tools.

centre-radiustangentnormalchord
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Copyright-safe content: These notes are rewritten from scratch. The uploaded PDFs were used only to understand chapter coverage, difficulty levels and test formats.

1. Introduction & Exam Weightage

Equation, tangent, normal, chord of contact and family of circles are frequent JEE coordinate tools.

Material signal: Mapped from straight-line and circle revision files.

Priority: High Yield. Treat this as a moderate chapter inside the JEE Mathematics ladder.

2. Core Concepts & Definitions

  • Complete squares to find centre and radius.
  • Tangent length uses power of a point.
  • Circle-line intersection is controlled by perpendicular distance from centre to line.

3. Key Formulas with Derivation Hints

  • (x-h)^2+(y-k)^2=r^2
    Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.
  • x^2+y^2+2gx+2fy+c=0
    Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.
  • centre=(-g,-f)
    Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.
  • radius=sqrt{g^2+f^2-c}
    Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.
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4. Solved Examples — Original Practice Models

Circle: main-style warm-up

Problem: A direct one-step Circle question is solved by identifying the governing definition before substituting values.

Method: write the target quantity, list the known values, use the standard relation, and check if the answer is dimensionally or logically possible. This prevents option-led guessing.

Circle: advanced-style trap

Problem: A multi-condition Circle problem often has an extra restriction hidden in domain, sign, interval or geometry.

Method: solve algebraically first, then filter using the domain/interval/geometry condition. In JEE Advanced, most wrong options come from skipping this final filtering step.

5. Common Mistakes & Traps

  • Missing factor 2 in general equation.
  • Taking radius squared as radius.
  • Ignoring imaginary circle condition.

6. JEE Main Specific Strategy

For JEE Main, aim for fast recognition and clean substitution. Finish the first pass of Circle questions in 60–90 seconds each. Prioritise standard formulas, short sign/domain checks and option elimination only after the setup is correct.

7. JEE Advanced Specific Strategy

For JEE Advanced, expect combined conditions, hidden domains, multi-correct traps and integer-style answers. Build the solution from definitions, not memorised tricks. When a parameter appears, solve the general case and then filter using restrictions.

8. Quick Revision Summary

  • Complete the square.
  • Compare distance with radius.
  • Power of point: S1.
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