Circles

Why This Chapter Matters

Circles is one of the favourite chapters for CBSE board exam questions — 5-8 marks every year. Tangent theorems (especially "tangent from external point") are asked in every exam. Proofs are compulsory for Long Answer questions.

Prerequisites

Core Concepts

1. Tangent to a Circle

A tangent to a circle is a line that touches the circle at exactly one point.

The point where the tangent meets the circle is called the point of tangency (or point of contact).

Number of tangents from different positions:

2. Theorem 1 — Tangent ⊥ Radius (Most Important!)

Statement: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

If O is the centre, P is the point of contact, and TP is the tangent:

OP ⊥ TP (OP is perpendicular to TP)

Proof outline:

3. Theorem 2 — Tangent from External Point

Statement: The lengths of the two tangents drawn from an external point to a circle are equal.

If PA and PB are tangents from external point P to circle with centre O:

PA = PB

Proof:

In △OAP and △OBP:

→ △OAP ≅ △OBP (RHS)

→ PA = PB □

Additional results from this proof:

Solved Examples

Example 1

Q: A tangent PQ at point P of a circle of radius 5 cm meets a line through centre O at Q, such that OQ = 13 cm. Find PQ.

OQ = 13, OP = 5 (radius), ∠OPQ = 90°

By Pythagoras: PQ = √(OQ² − OP²) = √(169 − 25) = √144 = 12 cm

Example 2

Q: Two concentric circles with radii 5 cm and 3 cm. Find the length of chord of larger circle that is tangent to the smaller circle.

Let chord AB of larger circle be tangent to smaller circle at P.

OP ⊥ AB (radius to tangent) → OP = 3 cm, OA = 5 cm

AP = √(25 − 9) = 4 cm

AB = 2 × AP = 8 cm

Example 3 — Classic Perimeter Problem

Q: From external point A, tangents AB and AC are drawn to a circle. BC is a chord. If AB = 4 cm, find perimeter of △ABC.

By tangent theorem: DB = DF and EC = EF (tangent from same external points D, E on BC)

Perimeter = AB + BC + AC = AB + BD + DC + AC = AB + BF + CF + AC = AB + AC + AC + AB...

Simpler version: Perimeter = 2 × (length of tangent) = 2 × 4 = 8 cm

PYQs

2023

Q: In figure, PQ is tangent to circle with centre O at Q. If ∠PQO = x° and ∠QPO = y°, prove that x − y = 90°.

∠OQP = 90° (radius ⊥ tangent), in △OQP: ∠QOP + x + y = 180°, but also x = 90° + y (exterior angle), so x − y = 90°

2022

Q: Prove that tangent to a circle is perpendicular to the radius at point of contact.

(Full formal proof required)

2021

Q: From external point P, two tangents PA and PB are drawn. O is centre. Prove that AB ⊥ OP.

△OAP ≅ △OBP (proved), so OP bisects ∠APB. Using this, show AB ⊥ OP

2020

Q: Two tangents TP and TQ are drawn from external point T. Prove TP = TQ and ∠PTQ = 2∠OPQ.

(Standard proof + angle relationship)

MCQ Practice

Q1. Number of tangents that can be drawn to a circle from a point inside it:

(A) 0 ✓ (B) 1 (C) 2 (D) Infinite

Q2. If tangent from external point has length 8 cm and radius is 6 cm, distance from external point to centre:

(A) 5 cm (B) 10 ✓ (C) 14 (D) √28

[d² = 8² + 6² = 100 → d = 10]

Q3 (Hard). AB is chord of circle with centre O. P is external point. PA and PB are tangents. Show ∠APB + ∠AOB = 180°.

∠OAP = ∠OBP = 90°, so in quadrilateral OAPB: ∠APB + ∠AOB = 360° − 180° = 180°

Revision Notes

KEY THEOREMS:
1. Tangent ⊥ Radius at point of contact
2. Tangents from external point are equal in length

Number of tangents:
  Inside circle → 0
  On circle → 1
  Outside circle → 2

For tangent problems, look for right angles (radius ⊥ tangent)
Then apply Pythagoras!

∠OPA = ∠OPB (OP bisects angle between tangents from P)

Common Mistakes:

❌ Assuming all chords bisect each other — only diameters do

❌ Forgetting ∠OAP = 90° when PA is tangent and OA is radius

❌ Using equal tangent theorem without justification in proof

Related Topics