Conic Sections — Circle

• The standard form of a circle and how to read off centre and radius.
• Converting between standard form (x−h)²+(y−k)²=r² and general form x²+y²+2gx+2fy+c=0.
• Tangent conditions to a circle from a point and the length of the tangent.
• The concept of power of a point with respect to a circle.

Level 1 — Standard and general form

Standard form: (x − h)² + (y − k)² = r²; centre (h, k), radius r.

Origin-centred: x² + y² = r².

General form: x² + y² + 2gx + 2fy + c = 0; centre (−g, −f), radius = √(g² + f² − c) (real circle requires g² + f² − c > 0).

Diametrically opposite points (x₁,y₁) and (x₂,y₂): Equation — (x−x₁)(x−x₂) + (y−y₁)(y−y₂) = 0.

Level 2 — Tangent and power of a point

Tangent from external point: Two tangents from external point P to circle; length of each = √(power of P). The pair of tangents forms an isoceles triangle with the chord of contact.

JEE tip: Always convert general form to standard form first by completing the square in both x and y.

NCERT spotlight — Completing the square

Convert x² + y² − 6x + 8y + 9 = 0 to standard form: Group x terms: (x² − 6x + 9) + (y² + 8y + 16) = 9 − 9 + 16 → (x−3)² + (y+4)² = 16. Centre (3, −4), radius 4.

Position of point: (x₁−h)² + (y₁−k)² compared to r². Less → inside; equal → on; greater → outside. Tangent exists only from outside point.

Intersection of line and circle: Substitute line into circle equation → quadratic in one variable. Discriminant D > 0 (secant), D = 0 (tangent), D < 0 (no intersection).

Find the equation of a circle passing through (1, 0), (0, 1), and (2, 2).

Step 1 — General form: x² + y² + 2gx + 2fy + c = 0. Three unknowns, three points.
Step 2 — Through (1, 0): 1 + 0 + 2g + 0 + c = 0 → 2g + c = −1 … (i)
Step 3 — Through (0, 1): 0 + 1 + 0 + 2f + c = 0 → 2f + c = −1 … (ii)
Step 4 — Through (2, 2): 4 + 4 + 4g + 4f + c = 0 → 4g + 4f + c = −8 … (iii)
Step 5 — (i) and (ii): 2g = 2f → g = f.
Step 6 — Substitute g = f in (i): 2g + c = −1.
Step 7 — (iii): 4g + 4g + c = −8 → 8g + c = −8.
Step 8 — Subtract: 6g = −7 → g = −7/6, f = −7/6, c = −1 − 2(−7/6) = −1 + 7/3 = 4/3.
Step 9 — Equation: x² + y² − (7/3)x − (7/3)y + 4/3 = 0.
   Multiply by 3: 3x² + 3y² − 7x − 7y + 4 = 0.

Applications — GPS circles and signal intersection

Two transmitters at known positions emit signals arriving with time difference → two circles; intersection gives position (basic GPS idea). Optical lenses use circular cross-sections; mirror curvature is circular near the axis. Engineering — pipe cross-sections, wheel design.

• Find centre and radius of x² + y² − 4x + 6y − 12 = 0.
• Write the tangent to x² + y² = 25 at point (3, 4).
• Find the length of tangent from (5, 0) to x² + y² = 9.

Open the Practice tab for graded questions on Circle.

• Use the platform-native live simulation or PhET-style tool for this topic (number line, Venn, physics playground, molecule builder, sensor dashboard, etc.).
• Mirror / body / home activity: physically do the concept (count objects, measure, role-play) and photograph or describe for portfolio.
• Voice or text reflection with AI Mentor: explain the concept to a younger student or family member.

• "Explain this concept to a Class 6 student using one real example from an Indian home, school, market, or festival."
• "What is one common mistake students make here, and how would you catch yourself making it?"
• Stretch: "How does this connect to coding, robotics, money, health, environment, or a future career?"

• Complete the core practice + one extension activity (photo, table, short reflection, or mini-project) for base XP + topic badge.
• 5-7 day streak or family discussion note = multiplier + visible artifact in parent/principal dashboard.
• Best real-world application stories (anonymised) featured on class or national leaderboard.

• One hands-on project or measurement using the Drishti kit or household items that makes the concept physical.
• Direct link to at least one Future Skill track (Money Management, Green Tech, Cyber Defenders, Micro-Entrepreneurship, AI Mastery, Sustainable Living, Personality Development).
• Coding extension where relevant (simple script, simulation, or data logging).

This material emphasises experiential "learning by doing", competency (apply/create/analyse), vocational exposure, critical thinking, and multidisciplinary connections. Designed to feed live worlds, AI Mentor (with memory), gamification, robotics, parent analytics, and future skills — not just exam prep.

Portfolio Evidence Idea: Your photo/table/reflection/project + one sentence on "How this helps me in real life or a possible future path."

Open the Practice tab for aligned questions (easy/medium/hard + case-based) with full AI scaffolding.

See curriculum for cross-links and the full future-skills/robotics chapters.