Class 10 Maths — Chapter 10: Circles

By RAAM SETU | NEET, SSC, Railway & Police Bharti Focused Notes 📘

📘 Introduction

This chapter deepens your understanding of Circles — one of the most important topics in Class 10 Mathematics. You’ll learn about tangents, secants, radii, chords, and circle theorems. These concepts are not only vital for board exams but also play a key role in NEET, SSC, Railway, and Police Bharti mathematics preparation.

🔹 Key Concepts & Theorems

📊 Chapter Summary Table

Concept Formula / Property Use
Tangent-Radius⊥ at point of contactHelps find angles & geometry relations
Equal TangentsPA = PBUsed in geometric proofs
Secant-Tangent RelationPT² = PQ × PRSolves problems involving lengths

🧮 Practice Questions

  1. Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
  2. Two tangents are drawn to a circle from an external point. Prove that they are equal in length.
  3. If a tangent and a secant are drawn from an external point, prove that the square of the tangent length is equal to the product of the secant’s external part and total length.

Circles — Class 10 (Advanced Notes)

Tangent & secant properties • Angles in a circle • Chords • Cyclic quadrilaterals • Equation of a circle • Practice
Jump to Practice

1. Definitions & Terminology

A circle is the set of all points in a plane at a fixed distance r (radius) from a fixed point O (centre). Key terms:

  • Radius: segment from centre to a point on circle.
  • Diameter: longest chord passing through centre = 2r.
  • Chord: a segment joining two points on circle.
  • Arc: part of circumference between two points.
  • Sector & Segment: sector = region bounded by two radii and arc; segment = region between chord and arc.

2. Angles in a Circle

  • Angle subtended by an arc at centre is twice the angle subtended at any point on the remaining part of the circle: ∠AOB = 2∠ACB.
  • Angle in a semicircle: angle subtended by diameter at any point on circle = 90°.
  • Angles in same segment: angles subtended by the same chord in the same segment are equal.

Use: To prove perpendicularity or equal angles when chords or arcs are given.

3. Chords & Perpendiculars

  • Equal chords subtend equal angles at centre and are equidistant from the centre.
  • Perpendicular from centre to a chord bisects the chord.
  • If two chords are equal, their corresponding arcs and subtended angles are equal.

4. Tangent & Secant Properties

  • Tangent-radius theorem: tangent at point P is perpendicular to radius OP.
  • Angle between tangent and chord: equals angle in alternate segment.
  • Two tangents from an external point: tangents from same external point are equal in length; external angle properties follow.
  • Power of a point (chord-chord & secant-tangent): If two chord segments meet at external point, product relations hold: PA × PB = PC × PD etc.

5. Cyclic Quadrilaterals

A quadrilateral is cyclic if all its vertices lie on a circle. Key property:

Opposite angles of a cyclic quadrilateral are supplementary ⇒ ∠A + ∠C = 180°, ∠B + ∠D = 180°.

Use cyclicity to relate angles and prove concyclicity from angle equalities.

6. Equation of a Circle (Coordinate Geometry)

Standard form with centre (h,k) and radius r:

(x − h)² + (y − k)² = r²

Special case: circle with centre at origin: x² + y² = r². Expand and compare coefficients to find centre and radius from general equation x² + y² + 2gx + 2fy + c = 0 where centre = (−g, −f) and r = √(g² + f² − c).

7. Solved Examples

Example 1. Prove that angle in a semicircle is 90° using isosceles triangles formed by radii.

Example 2. If two tangents from external point P touch circle at A and B, show PA = PB and ∠APO = 90° − (1/2)∠AOB where O is centre.

Example 3. Find centre and radius of circle x² + y² − 4x + 6y − 12 = 0. Answer: centre (2, −3), r = √(4 + 9 + 12) = √25 = 5.

8. Practice Questions

  1. Prove that the angle between tangent and radius is 90°.
  2. Find equation of circle with centre (3,2) and radius 7.
  3. Show that if a cyclic quadrilateral has three right angles then the fourth is also right angle.
  4. Two chords AB and CD intersect at P inside circle such that AP = 3, PB = 4, CP = 2. Find PD using chord product property.

9. Exam Tips & Strategy

  • Draw accurate diagrams and mark equal radii; most proofs are angle chasing + isosceles triangle facts.
  • Use chord and tangent properties to convert geometry problems into algebraic relations quickly.
  • For coordinate problems, complete square to find centre and radius from general equation.
  • Memorise key theorems: angle in semicircle, tangent–radius, equal tangents, cyclic opposite angles.