Circles Class 11: Complete Guide, All Formulas & Free PDF Download (JEE & Boards)
Circles — Competishun
Circles Class 11: Complete Guide, All Formulas & Free PDF Download (JEE & Boards)
Circles (Coordinate Geometry) is one of the most fundamental and scoring chapters in Class 11 Maths. It carries significant weightage in JEE and is a guaranteed source of marks in board exams. The chapter deals with the study of circles in the coordinate plane, which is essential for understanding conic sections.
This chapter covers standard and general equations of a circle, tangent and normal, chord properties, family of circles, and power of a point. A circle is the set of all points in a plane that are at a fixed distance (radius) from a fixed point (center).
This page gives you the complete guide to Circles with all concepts explained in depth. You will find clear definitions, formulas, visual diagrams, and practice questions. Download the free PDF below and keep it handy for quick revision before your JEE Main, board exams, or any other test.
Download the Circles Complete Guide PDF
Get all Circles concepts, standard and general equations, tangent, normal, chord, family of circles, and power of a point in one clean PDF, free. Perfect for JEE and board revision.
Download Free PDFWhat is a Circle in Coordinate Geometry?
A circle is the simplest conic section. It is the locus of a point that moves such that its distance from a fixed point is constant.[reference:1]
Center
The fixed point from which every point on the circle is equidistant. Denoted by (h, k).
Radius
The fixed distance from the center to any point on the circle. Denoted by r.
Glossary of Circles Terms — Complete A to Z
Before diving deep into each topic, let's understand the key terminology used in this chapter:
| Term | Definition |
|---|---|
| Circle | The set of all points in a plane at a fixed distance from a fixed point.[reference:2] |
| Center | The fixed point from which all points on the circle are equidistant. |
| Radius (r) | The distance from the center to any point on the circle.[reference:3] |
| Diameter | A chord passing through the center; it is twice the radius.[reference:4] |
| Chord | A line segment joining any two points on the circle.[reference:5] |
| Tangent | A line that touches the circle at exactly one point.[reference:6] |
| Normal | A line perpendicular to the tangent at the point of contact.[reference:7] |
| Arc | A part of the circumference of a circle. |
| Sector | The region bounded by two radii and the arc between them. |
| Segment | The region bounded by a chord and the arc between its endpoints. |
| Secant | A line that intersects the circle at two points. |
| Concentric Circles | Circles with the same center but different radii. |
| Cyclic Quadrilateral | A quadrilateral whose vertices all lie on a single circle.[reference:8] |
| Power of a Point | A measure of the distance of a point from a circle.[reference:9] |
| Director Circle | The locus of points from which tangents to a circle are perpendicular.[reference:10] |
| Mastering these terms is essential for understanding Circles. | |
Standard Equation of a Circle
(x − h)² + (y − k)² = r²[reference:11][reference:12]
Special Cases
General Equation of a Circle
x² + y² + 2gx + 2fy + c = 0[reference:13][reference:14]
where the center is (−g, −f) and the radius is √(g² + f² − c)[reference:15][reference:16]
| Parameter | Value | Explanation |
|---|---|---|
| Center | (−g, −f) | From the general equation |
| Radius | √(g² + f² − c) | Must be positive for a real circle |
| Condition for Real Circle | g² + f² − c > 0 | Otherwise, the equation represents a point or imaginary circle |
| The general equation is the most versatile form for solving problems. | ||
Diameter Form of a Circle
(x − x₁)(x − x₂) + (y − y₁)(y − y₂) = 0[reference:17]
(x − 1)(x − 3) + (y − 2)(y − 4) = 0
⇒ x² + y² − 4x − 6y + 11 = 0
Parametric Equations of a Circle
x = h + r cos θ
y = k + r sin θ
where 0 ≤ θ ≤ 2π[reference:18]
- Parametric point: Any point on the circle can be represented as (h + r cos θ, k + r sin θ).
- θ is the parameter: It represents the angle made by the radius with the positive x-axis.
Tangent and Normal to a Circle
Tangent to a Circle
Circle: x² + y² = r²
Circle: x² + y² = r²
Circle: x² + y² = r²
General Circle
Normal to a Circle
Circle: x² + y² = r²
General Circle
Condition for Tangency
The line y = mx + c is tangent to the circle x² + y² = r² if:
The distance from the center to the line must equal the radius.[reference:27]
Chord Properties of a Circle
Important Chord Properties
Perpendicular from Center
Intersecting Chords
Length of Chord
Chord of Contact
Power of a Point
P = (x₁ − h)² + (y₁ − k)² − r²[reference:31]
| Value of Power | Position of Point |
|---|---|
| P > 0 | Point is outside the circle[reference:32] |
| P = 0 | Point is on the circle[reference:33] |
| P < 0 | Point is inside the circle[reference:34] |
| The power of a point is a fundamental concept in circle geometry. | |
Family of Circles
Passing through intersection of S=0 and S'=0
Passing through intersection of S=0 and line L=0
Touching S=0 at P with common tangent L=0
Having a fixed center
Passing through a fixed point
Practice Questions — From JEE and Boards
| Question | Answer |
|---|---|
| Q1: Find the equation of a circle with center (2, -3) and radius 5. | (x − 2)² + (y + 3)² = 25. |
| Q2: Find the center and radius of the circle x² + y² − 4x + 6y − 12 = 0. | Center: (2, −3), Radius: √(4 + 9 + 12) = 5. |
| Q3: Find the equation of the tangent to x² + y² = 16 at (4, 0). | xx₁ + yy₁ = r² ⇒ 4x = 16 ⇒ x = 4. |
| Q4: Find the length of the tangent from (6, 8) to the circle x² + y² = 25. | Length = √(36 + 64 − 25) = √75 = 5√3. |
| Q5: Find the equation of the circle with diameter joining (1, 2) and (3, 4). | (x − 1)(x − 3) + (y − 2)(y − 4) = 0 ⇒ x² + y² − 4x − 6y + 11 = 0. |
| Q6: For what value of k does the line y = 2x + k touch the circle x² + y² = 5? | Condition: c = ± r√(1 + m²) ⇒ k = ± √5 · √(1 + 4) = ± 5. |
| Q7: Find the equation of the family of circles passing through the intersection of x² + y² = 4 and x² + y² − 2x − 2y = 0. | S₁ + λS₂ = 0 ⇒ x² + y² − 4 + λ(x² + y² − 2x − 2y) = 0. |
| Q8: Find the power of the point (1, 2) with respect to the circle x² + y² = 9. | P = 1 + 4 − 9 = −4 (point is inside the circle). |
| Practise these types of questions to become comfortable with applying Circles concepts in exam scenarios. | |
All Circles Formulas at a Glance
| Formula | What It Means |
|---|---|
| (x − h)² + (y − k)² = r² | Standard equation of a circle[reference:44] |
| x² + y² + 2gx + 2fy + c = 0 | General equation of a circle[reference:45] |
| Center: (−g, −f) | Center from general equation[reference:46] |
| Radius: √(g² + f² − c) | Radius from general equation[reference:47] |
| (x − x₁)(x − x₂) + (y − y₁)(y − y₂) = 0 | Diameter form[reference:48] |
| x = h + r cos θ, y = k + r sin θ | Parametric equations[reference:49] |
| xx₁ + yy₁ = r² | Tangent at (x₁, y₁)[reference:50] |
| y = mx ± r√(1 + m²) | Tangent in slope form[reference:51] |
| c = ± r√(1 + m²) | Condition for tangency[reference:52] |
| Power of point: (x₁ − h)² + (y₁ − k)² − r² | Power of a point[reference:53] |
| S + λS' = 0 | Family of circles[reference:54] |
| Memorise these formulas for Circles. They are the key to scoring full marks in this chapter. | |
Common Mistakes in Circles
- Confusing the general equation coefficients: In x² + y² + 2gx + 2fy + c = 0, the center is (−g, −f), not (g, f).
- Forgetting the condition for a real circle: g² + f² − c must be positive. Otherwise, the equation does not represent a real circle.
- Misapplying the tangent formula: For x² + y² = r², the tangent at (x₁, y₁) is xx₁ + yy₁ = r², not x/x₁ + y/y₁ = 1.
- Confusing tangent and normal: The normal passes through the center; the tangent is perpendicular to the radius.
- Forgetting the sign in the power of a point: Power is (x₁ − h)² + (y₁ − k)² − r². If it's negative, the point is inside the circle.
- Not checking the condition for tangency of a line: The line y = mx + c is tangent if c = ± r√(1 + m²).
Why Circles Matters for JEE and Boards
- Foundation for conic sections: Circles are the simplest conic section and form the basis for understanding ellipses, parabolas, and hyperbolas.
- High weightage: This chapter appears in 1-2 questions in JEE Main and is a guaranteed source of marks in board exams.
- Conceptual clarity: This chapter rewards students who understand the definitions and properties rather than just memorizing formulas.
- Practical relevance: Circles are used everywhere, from physics to engineering to computer graphics.
Get the Complete Circles PDF for Free
Download the full Circles guide with all concepts, definitions, formulas, and practice questions. Perfect for last-minute revision before JEE and board exams.
Download Free PDFMaster Class 11 Maths with Competishun
Competishun teaches complete JEE mathematics with structured lectures, DPPs, PYQs and an All India Test Series.
All taught by faculty with 18 to 25 years of experience.
Pratham
Two-year foundation program that builds strong Class 11 maths, physics and chemistry concepts.
Explore PrathamPrakhar
Class 12 program balancing boards, JEE Main and Advanced preparation together.
Explore PrakharPraveen & Fast Lane
Dropper program for a focused repeat attempt, including the accelerated Fast Lane batch.
Explore PraveenPragyaan
Live interactive batch with real-time classes and mentorship for JEE and NEET.
Explore PragyaanDLP Study Material
Complete printed material with DPPs and PYQs for Class 11, Class 12 and droppers, delivered home.
Explore DLPAll India Test Series
Regular JEE and NEET level tests with detailed analysis to track and improve your rank.
Explore Test SeriesFree Maths Lectures on YouTube
Want to see Circles taught step by step with clear explanations? Competishun's full concept lectures are free on its YouTube channel.
More than 2.2 million+ students follow the channel for free concept lectures, PYQ solutions and exam strategy.
Learn the concepts, then use this Circles guide to revise.
Build Your Maths Foundation
Class 11, Class 12 or dropper, explore every Competishun program.
Or start free with full concept lectures on YouTube.
About Competishun
Competishun is an online JEE and NEET coaching platform with programs for Class 11, Class 12 and droppers.
Complete DLP study material and an All India Test Series covering JEE Main, Advanced and NEET.
Taught by faculty with 18 to 25 years of experience.
More than 2.2 million+ students follow the Competishun YouTube channel.
Free JEE and Boards Concept Lectures
Follow Competishun for free Physics, Chemistry and Maths lectures, PYQ solutions and strategy.
Read These Next
Types of matrices, operations, determinant properties, adjoint, inverse, and solving linear equations explained in depth.
Domain, range, principal values, properties, and formulas of inverse trigonometric functions explained in depth.
AP, GP, HP, AM, GM, HM, and infinite series explained in depth.
Frequently Asked Questions — Circles
Tags
