AdvanceReview
…Checking

Math notesCoordinateCircle

Independent topic page

Circle

Circle full marks: complete square → centre/radius → one tool (tangent length, chord of contact, S₁−S₂) → verify with a point.

Portion CoordinateFocus High join with lines and conicsTarget 60–90sPortion share 12.8%Trend Falling share
Syllabus: circle forms, tangent/normal/chord, parametric form, intersections with line/circle, circle through intersection points. Standardise before any formula. Circle Advanced starts with centre/radius hygiene — then the dictionary pays off.

How Advanced asks this

  • Tangent from a point / length of tangent
  • Chord of contact / pole-polar style
  • Common chord of two circles
  • Circle–line intersection / tangency condition
  • Family of circles through intersection

Traps that leak marks

  • Wrong radius after completing square
  • g,f sign flip
  • √S₁ when point inside
  • Expanding both circles instead of S₁−S₂
  • √S₁ as length from an interior point
  • Forgetting complete-square before reading centre
  • Chord of contact with a miswritten T

01. General x²+y²+2gx+2fy+c=0 → centre (−g,−f), r=√(g²+f²−c). Completing square prevents g,f sign errors.

02. Length of tangent from (x₁,y₁) is √S₁ (when outside). Chord of contact from external point is T=0.

03. Two circles: S₁−S₂=0 is radical axis / common chord line when they intersect.

04. Condition that y=mx+c touches x²+y²=r²: c²=r²(1+m²) (standard); generalise carefully after shift.

05. Advanced lane: general equation → complete the square before centre/radius tools. Wrong centre poisons every later chord/tangent formula.

06. Two circles: S₁−S₂ gives the common chord line; power/length tools beat solving simultaneous quadratics when the ask is chord or length.

Official · 2026

Syllabus for this topic

Official syllabus points for Circle only — not the full Coordinate portion map. Portion hub: Coordinate · Coordinate Geometry analysis.

Syllabus on this page is topic-scoped: Circle. The parent portion (Coordinate Geometry) also covers other topics — use the portion hub for the full map.

Must cover (this topic)

  • Circle tangent/normal/chord + parametric

Gaps / watch-outs

No topic-specific exclusions flagged — finish the unit points below for Circle.

Source: https://jeeadv.ac.in/documents/jee-advanced-2026-syllabus.pdf

Unit 01

Circle

high

Exam focusCircle equations, chords, and tangents under Advanced ask types.

Syllabus points

  • Circle: forms, tangent/normal/chord, parametric; intersections

Full Mathematics syllabus →

Cross-topic stems

Mixed joins for this topic

Advanced rarely asks a chapter in isolation. These are tagged multi-idea Math stems that touch Circle — practise the knowledge bridge, not only the single-chapter drill.

Mathematics2026 · Paper 1 · Q2single-correct

Parabola points + circumcircle + ellipse

Coordinate Geometry – Parabola+Circle (circumradius)+Ellipse conditions

Knowledge bridge

Step 1: recover points P, Q, R from the slope/tangent conditions on the parabola. Step 2: find the circumcircle of triangle PQR (perpendicular bisectors). Step 3: read the required radius / relation that also ties to an ellipse option.

Why it feels hard

Geometry hops across three curve types in one SC item.

What to practise

Practise ‘points on parabola → circle through them → conic check’ as a single drill.

2026 year review →Mathematics notes →Quick card →Syllabus →
Mathematics2026 · Paper 1 · Q16matching

Matching across circle / parabola / ellipse / hyperbola

Circle+Parabola+Ellipse+Hyperbola

Knowledge bridge

Step 1: for each List-I geometric condition, reduce to a standard conic equation or point. Step 2: match coordinates/parameters in List-II without mixing ellipse and hyperbola templates. Step 3: verify one substituted point per match to avoid swap errors.

Why it feels hard

Matching lists punish template mix-ups under time pressure.

What to practise

Keep a one-page conics identity sheet and practise full matching sets weekly.

2026 year review →Mathematics notes →Quick card →Syllabus →

All mixed joins →

Why methods matter

Common methods that change your score

Circle is standardise-then-tool. These methods stop g,f sign errors and force S₁−S₂ for two-circle stems — the usual time sinks.

01

Complete square first

Removes the #1 radius/centre error before any tool is applied.

Do thisNever apply focus-style conic memory on a general circle without centre/r.

Why it works

Removes the #1 radius error.

See it once

x²+y²−4x+6y−3=0 → (x−2)²+(y+3)²=16 → r=4.

02

S₁−S₂ shortcut

Two-circle chords become a line + Pythagoras — not a quadratic jungle.

Do thisFor intersection problems, subtract equations early.

Why it works

Linear common chord without solving quadratics fully.

See it once

Then drop perpendicular from centre to get chord length.

03

Position test

Stops illegal √S₁ from interior points.

Do thisEvaluate S₁ sign before tangent-length formula.

Why it works

Prevents illegal √ of negative.

See it once

S₁>0 outside, =0 on, <0 inside (for S=0 normalised form).

04

T=0 discipline

Chord of contact errors are mostly T miswrites — this fixes the habit.

Do thisWrite T correctly for the given circle form.

Why it works

Chord of contact errors are mostly T miswrites.

See it once

For x²+y²=r², T is xx₁+yy₁=r².

Processed examples

Sample questions · full working

Read the impact line first. Then notice which family/model was chosen. The solution is written the way your rough sheet should look in the hall.

01

bankFamily · Complete square → tool

Sample question

Length of tangent from (4,0) to x²+y²=4.

√S₁ is instant once you know the point is outside.

Model chosenS₁ at (4,0) → √S₁.

Processed solution

  1. For x²+y²−4=0, evaluate S₁ at (4,0): 16−4=12.
  2. S₁>0 ⇒ point outside the circle.
  3. Tangent length = √S₁ = √12 = 2√3.

Answer2√3

Takeaway: Check S₁ sign before taking square root.

02

standardFamily · Complete square → tool

Sample question

Find centre and radius of x²+y²−4x+6y−3=0.

Completing square (or reading g,f,c) is the gate for every later tool.

Model chosenComplete square / centre (−g,−f).

Processed solution

  1. Complete the square: (x−2)²+(y+3)² −4 −9 −3 = 0.
  2. (x−2)²+(y+3)² = 16.
  3. Centre (2,−3), radius 4.

AnswerCentre (2,−3), r=4

Takeaway: Standardise before any formula sheet.

03

twistFamily · Common chord / radical axis

Sample question

Find the length of the common chord of x² + y² − 4x − 5 = 0 and x² + y² − 4y − 5 = 0.

S₁ − S₂ gives the chord without solving two quadratics — that is the scoring move.

Model chosenRadical axis → centre-to-chord distance → half-chord.

Processed solution

  1. Subtract: −4x + 4y = 0 ⇒ y = x (common chord).
  2. First circle: centre (2, 0), radius 3.
  3. Distance from (2, 0) to x − y = 0 is |2|/√2 = √2.
  4. Half-chord = √(9 − 2) = √7 ⇒ full length 2√7.

Answer2√7

Takeaway: Common chord ⇒ subtract circles first; never chase both intersections.

Reusable models

Common question families

Advanced stems cluster. Name the family in 10–15s, write the fast model, then run the same path. Do not invent a new plan each time.

01

Complete square → tool

Looks likeGeneral circle; tangent length / power / centre

Fast modelCentre (−g,−f), r → then √S₁ or T=0.

Same solve path

  1. Standardise.
  2. Write centre, r.
  3. Apply one tool.
  4. Position check (inside/out).

Rough skeleton

  • Centre,r
  • Tool
  • Answer

Why faster

Standardisation unlocks every dictionary formula.

Refuse this

Do not apply √S₁ before knowing the circle form.

02

Common chord / radical axis

Looks likeTwo circles; common chord or length

Fast modelS₁−S₂=0 immediately; then perpendicular from centre + Pythagoras.

Same solve path

  1. S₁−S₂ line.
  2. Distance from centre.
  3. Half-chord √(r²−d²).
  4. Double if full length.

Rough skeleton

  • S₁−S₂
  • d
  • Length

Why faster

Avoids solving the full quadratic system.

Refuse this

Do not expand both circles fully entry-wise first.

03

Line–circle tangency

Looks likeCondition that a line touches the circle; find c or m

Fast modelDistance from centre to line = r (or standard c²=r²(1+m²) after shift).

Same solve path

  1. Centre, r.
  2. Distance = r condition.
  3. Solve parameter.

Rough skeleton

  • Centre
  • dist=r
  • Param

Why faster

One geometric condition replaces substitution into quadratic discriminant.

Refuse this

Do not always solve intersection quadratic if distance=r is enough.

04

Orthogonal / director / power MC

Looks likeOrthogonal circles; power of a point; angle between circles

Fast model2g₁g₂+2f₁f₂=c₁+c₂ for orthogonal; power = S₁ at the point.

Same solve path

  1. Standardise both circles.
  2. Apply orthogonality or power formula.
  3. Solve parameter / read value.

Rough skeleton

  • Forms
  • Condition
  • Answer

Why faster

Named conditions kill options without intersection algebra.

Refuse this

Do not find both intersection points to test orthogonality.

Hall timing

60–90s process

  1. 0–15s
    Standardise to centre-radius or read g,f,c.

    Write centre and r on rough sheet before tools.

  2. 15–35s
    Pick tool: √S₁ / T=0 / S₁−S₂ / tangency condition.

    If two circles, prefer radical axis before expanding both fully.

  3. 35–75s
    Execute algebra; keep r under square root non-negative.

    For common chord length: perpendicular from centre to chord + Pythagoras.

  4. 75–90s
    Check point position (inside/on/outside) matches formula use.

    S₁<0 inside — tangent length not real; don’t force √S₁.

After the model is locked

Quick formula bank

Standard / general

(x−h)²+(y−k)²=r²; x²+y²+2gx+2fy+c=0 → centre (−g,−f), r=√(g²+f²−c)

Always start here

Length of tangent

√S₁ from (x₁,y₁)

External point

Chord of contact

T=0 from (x₁,y₁)

Pair of tangents’ chord

Radical axis

S₁−S₂=0

Common chord / radical axis

Parametric

(h+r cosθ, k+r sinθ)

Point on circle

Line tangency (x²+y²=r²)

c²=r²(1+m²) for y=mx+c

Slope form tangents to centred circle

Power

PA·PB = power = |S₁| ideas

Intersecting chord theorems in coordinate form

Mental bank

  • Centre (−g,−f), r=√(g²+f²−c)
  • Tangent length √S₁
  • S₁−S₂ common chord
  • T=0 chord of contact
  • c²=r²(1+m²) for x²+y²=r²
  • Parametric (h+r cosθ, k+r sinθ)
  • x²+y²+2gx+2fy+c=0 → centre (−g,−f), r=√(g²+f²−c)
  • Tangent condition: distance from centre = r
  • T=0 for pole/polar and chord of contact
  • S₁−S₂ = common chord (for intersecting circles)

Make it automatic

Practice · checklist

Daily

  • 1 complete-square + tool.
  • 1 tangent length or T=0.
  • Alt days: 1 S₁−S₂ common chord.

Weekly

  • 1 circle–line tangency set.
  • 1 circle–conic mix.
  • Error log: centre sign errors.

100% checklist

  • Always standardise first.
  • S₁ sign checked before √S₁.
  • S₁−S₂ used for two-circle chords.
  • T written for correct form.
  • ≥90% on last timed Circle set.
  • Centre and r extracted via complete square when needed.
  • Position of point (inside/on/outside) checked before length formulae.
  • T written carefully for contact/chord-of-contact stems.
Coordinate hubAll Math notesSyllabusSubject analysisMixed joins