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Class 10 . Mathematics . Mathematics . Circles

Chapter 10: Circles

Ch 10

Mathematics

Class 10

Lecture 1

Introduction

Reference video · Lecture 1Introduction
Definition

Circle — A circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, called the centre.

A Line and a Circle

Possible Positions

  • No common point: The line is non-intersecting with respect to the circle.
  • Two common points: The line is a secant; the points on the circle are the endpoints of a chord.
  • One common point: The line is a tangent; the shared point is the point of contact.
Three positions of a line relative to a circle: outside, intersecting at two points, and tangent at one point.
Three positions of a line relative to a circle: outside, intersecting at two points, and tangent at one point.
Definition

Tangent — A tangent is a line that meets a circle at exactly one point.

Tangent Facts

  • Secant connection: A tangent is the limiting case of a secant when the two endpoints of the corresponding chord coincide.
  • At a point: Exactly one tangent can be drawn to a circle at any given point on it.
  • Parallel tangents: There can be at most two tangents to a circle parallel to a given secant.
  • Everyday model: A rope touching a pulley, or the ground touching a wheel, can represent a tangent.

Radius and Tangent

Tangent–Radius Theorem

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

$$OP \perp XY$$
Tangent–Radius Relationship
$$OP \perp XY \quad \text{and} \quad \angle OPQ = 90^\circ$$

Use when a line is tangent at point \(P\) to a circle with centre \(O\).

A circle with centre O and tangent XY at P; the radius OP is perpendicular to the tangent.
A circle with centre \(O\) and tangent \(XY\) at \(P\); the radius \(OP\) is perpendicular to the tangent.

Why the Radius Is Perpendicular to the Tangent

  1. Let \(XY\) be tangent to the circle at \(P\), with centre \(O\); choose any other point \(Q\) on \(XY\).
  2. Since \(Q\) is not the point of contact, it lies outside the circle, so its distance from \(O\) exceeds the radius.
    $$OQ > OP$$
  3. Thus \(OP\) is the shortest distance from \(O\) to the line \(XY\); the shortest distance from a point to a line is perpendicular to it.
    $$OP \perp XY$$

Related Term

  • Normal: The line containing the radius through the point of contact is called the normal to the circle at that point.
Worked example

A line \(XY\) touches a circle with centre \(O\) at \(P\). Find the angle between \(OP\) and \(XY\).

  1. A line \(XY\) touches a circle with centre \(O\) at \(P\). Find the angle between \(OP\) and \(XY\).
  2. The radius through the point of contact is perpendicular to the tangent.
    $$OP \perp XY$$
  3. Therefore, the angle between them is a right angle.
    $$90^\circ$$
Answer: The angle is \(90^\circ\).
Practice

A line cuts a circle at exactly two points. What is the line called?

Hint: Classify the line by its number of common points with the circle.
Key step: Two common points identify a secant.
Solution: It is a secant.
Practice

A tangent touches a circle at \(P\). If \(O\) is the centre, what is the angle between \(OP\) and the tangent?

Hint: Use the tangent–radius theorem.
Key step: The radius to the point of contact is perpendicular to the tangent.
Solution: The angle is \(90^\circ\).
Practice

What is the maximum number of tangents to a circle that can be parallel to a given secant?

Hint: Consider the two sides of the secant.
Key step: A parallel tangent can occur on either side of the secant.
Solution: At most two.

Quick recap

  • A line and a circle can have zero, one, or two common points.
  • A line with one common point is a tangent; that point is the point of contact.
  • A tangent is a limiting case of a secant when the chord endpoints coincide.
  • There is exactly one tangent at any given point of a circle.
  • The radius through the point of contact is perpendicular to the tangent.
  • The radius line through the point of contact is also called the normal.
Lecture 2

Number Of Tangents From A Point On A Circle

Reference video · Lecture 2Number Of Tangents From A Point On A Circle

How Many Tangents Can Be Drawn?

Definition

Number of tangents from a point — The number of tangents from a point depends on whether it lies inside, on, or outside the circle.

Point Position and Number of Tangents
Position of pointNumber of tangentsReason
Inside the circle0Every line through the point intersects the circle at two points.
On the circle1There is exactly one tangent at the point of contact.
Outside the circle2Exactly two tangents can be drawn from the point.
A point inside the circle has no tangent.
A point inside the circle has no tangent.
There is one tangent at a point on the circle.
There is one tangent at a point on the circle.
Two tangents from an external point touch the circle at two distinct points.
Two tangents from an external point touch the circle at two distinct points.

Equal Tangents from an External Point

Equal tangents theorem

The lengths of two tangents drawn from the same external point to a circle are equal. If tangents from \(P\) touch the circle at \(Q\) and \(R\), then \(PQ=PR\). Join the centre \(O\) to \(P\), \(Q\), and \(R\): \(OQ\) and \(OR\) are radii perpendicular to the tangents. Thus right triangles \(OQP\) and \(ORP\) are congruent by RHS, so \(PQ=PR\) by CPCT.

The two tangent segments from the same external point are equal.
The two tangent segments from the same external point are equal.
Tangent Length and Angle-Bisector Properties
$$PQ=PR,\qquad \angle QPO=\angle OPR$$

The line joining the external point to the centre bisects the angle between the two tangents.

Useful Consequences

  • Equal tangent lengths: Tangents from the same external point are equal.
  • Angle bisector: The centre-to-external-point line bisects the angle between the two tangents.
  • Right triangle: For an external point \(X\), centre \(O\), radius \(r\), and tangent length \(XT\), \(XT^2=OX^2-r^2\).

Solving Tangent-Length Problems

  1. Join the centre to the external point and to each point of contact.
  2. Use the perpendicular radius–tangent property to identify right triangles.
  3. Apply equal tangents, congruence, similarity, or Pythagoras as appropriate.
  4. Substitute the known lengths and solve for the required quantity.
Worked example

A chord \(PQ\) of length \(8\) cm in a circle of radius \(5\) cm has tangents at \(P\) and \(Q\) meeting at \(T\). Find \(TP\).

  1. A chord \(PQ\) of length \(8\) cm in a circle of radius \(5\) cm has tangents at \(P\) and \(Q\) meeting at \(T\). Find \(TP\).
  2. Since the tangents from \(T\) are equal, \(OT\) bisects the angle between them; hence it bisects chord \(PQ\) at \(R\) and is perpendicular to it.
    $$PR=RQ=\frac{8}{2}=4$$
  3. Use right triangle \(OPR\) to find the distance from the centre to the chord.
    $$OR=\sqrt{OP^2-PR^2}=\sqrt{5^2-4^2}=3$$
  4. Triangles \(TRP\) and \(PRO\) are similar, so their corresponding sides are proportional.
    $$\frac{TP}{PO}=\frac{PR}{RO},\qquad \frac{TP}{5}=\frac{4}{3}$$
  5. Solve for the tangent length.
    $$TP=\frac{20}{3}\text{ cm}$$
Answer: \(TP=\frac{20}{3}\) cm.

Applications

Chord of a Larger Circle Touching a Concentric Smaller Circle

  • Setup: Two circles share centre \(O\). A chord \(AB\) of the larger circle touches the smaller circle at \(P\).
  • Perpendicular: \(OP\) is a radius to the point of contact, so \(OP\perp AB\).
  • Conclusion: A perpendicular from the centre of a circle to a chord bisects the chord; therefore, \(AP=PB\).
Angle Between Two Tangents
$$\angle PTQ=2\angle OPQ$$

For tangents \(TP\) and \(TQ\) meeting at \(T\), \(\angle OPQ\) is the angle between \(OP\) and \(PQ\) (not between a radius and a tangent).

Why the Angle Relation Holds

  • Equal tangents: \(TP=TQ\), so triangle \(TPQ\) is isosceles.
  • Base angles: If \(\angle PTQ=\theta\), then \(\angle TPQ=90^\circ-\frac{\theta}{2}\).
  • Radius and tangent: \(OP\perp TP\), so \(\angle OPT=90^\circ\). Since \(\angle OPQ=\angle OPT-\angle TPQ\), \(\angle OPQ=\frac{\theta}{2}\) and thus \(\angle PTQ=2\angle OPQ\).
Practice

Two concentric circles have centre \(O\). A chord \(AB\) of the larger circle touches the smaller circle at \(P\). Prove that \(AP=PB\).

Hint: Join the centre to the point of contact.
Key step: Use the radius–tangent perpendicular property and the perpendicular-from-centre chord theorem.
Solution: Since \(OP\) is a radius to the contact point, \(OP\perp AB\). A perpendicular from the centre to chord \(AB\) bisects it. Therefore, \(AP=PB\).
Practice

Tangents \(TP\) and \(TQ\) are drawn from \(T\) to a circle with centre \(O\). Prove that \(\angle PTQ=2\angle OPQ\).

Hint: Use equal tangent lengths, then relate the base angle of isosceles triangle \(TPQ\) to the right angle between \(OP\) and \(TP\).
Key step: Set \(\angle PTQ=\theta\) and find \(\angle TPQ\).
Solution: Since \(TP=TQ\), \(\angle TPQ=90^\circ-\frac{\theta}{2}\). Also, \(OP\perp TP\), so \(\angle OPQ=90^\circ-\angle TPQ=\frac{\theta}{2}\). Hence, \(\angle PTQ=2\angle OPQ\).
⚠ Common mistake

The equal-tangents theorem applies only to tangents drawn from the same external point. Also, \(\angle OPQ\) is between \(OP\) and \(PQ\); the radius is perpendicular to tangent \(PT\), not to chord \(PQ\).

Quick recap

  • A point inside, on, or outside a circle has respectively \(0\), \(1\), or \(2\) tangents.
  • A tangent is perpendicular to the radius at the point of contact.
  • Two tangent segments from the same external point are equal.
  • The line from the centre to the external point bisects the angle between the tangents.
  • A perpendicular from the centre to a chord bisects the chord.
  • For tangents \(TP\) and \(TQ\), \(\angle PTQ=2\angle OPQ\) when \(P,Q\) are the contact points.

Formula sheet

Tangent–Radius Relationship
$$OP \perp XY \quad \text{and} \quad \angle OPQ = 90^\circ$$
Tangent Length and Angle-Bisector Properties
$$PQ=PR,\qquad \angle QPO=\angle OPR$$
Angle Between Two Tangents
$$\angle PTQ=2\angle OPQ$$

Mathematics — ALL CHAPTERS

1

Real Numbers

2

Polynomials

3

Pair Of Linear Equations In Two Variables

4

Quadratic Equations

5

Arithmetic Progressions

6

Triangles

7

Coordinate Geometry

8

Introduction To Trigonometry

9

Some Applications Of Trigonometry

10

Circles

11

Areas Related To Circles

12

Surface areas And Volumes

13

Statistics 

14

Probability