CBSE Class 10 Mathematics Circles Notes

About This Chapter

1. Introduction and Definition

A circle is the locus of all points in a plane that are equidistant from a fixed point called the centre. The fixed distance from the centre to any point on the circle is called the radius. This chapter focuses not on the circle itself but on lines that interact with it in specific ways.

1.1 Basic Circle Terminology

1.2 Number of Tangents from a Point

2. Key Concepts and Components

2.1 Tangent to a Circle

A tangent to a circle is a line that touches the circle at exactly one point. Unlike a secant, a tangent does not cross the circle; it merely grazes its boundary. The point at which the tangent meets the circle is called the point of contact or point of tangency.

2.2 Secant to a Circle

A secant is a line that intersects the circle at two distinct points. As the secant is moved closer and closer to become tangent to the circle, the two points of intersection converge into a single point of tangency. This is the limiting case of a secant.

2.3 Relationship Between Tangent and Radius

The most fundamental property of a tangent is the angle it makes with the radius at the point of tangency. This relationship is the foundation for virtually all circle theorems in this chapter.

2.4 Tangent from an External Point

When two tangents are drawn from an external point to a circle, they have equal length. This is a direct consequence of the congruence of the two right triangles formed by the tangents, the radii, and the line joining the external point to the centre.

3. Core Theorems and Derivations

3.1 Theorem 1: Tangent Perpendicular to Radius

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

Given: A circle with centre O, tangent XY at point P on the circle.

To Prove: OP is perpendicular to XY.

Proof: Let Q be any other point on the tangent line XY. Since XY is a tangent, Q lies outside the circle. Therefore OQ > OP (as OP = radius, and OQ is greater than the shortest distance from O to the line, which is OP since OP is perpendicular). Since this is true for every point Q on XY other than P, OP is the shortest distance from O to the line XY. Therefore OP is perpendicular to XY.

3.2 Theorem 2: Equal Tangents from External Point

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

Given: Circle with centre O. External point P. PA and PB are tangents touching at A and B.

To Prove: PA = PB

Proof: In triangles OAP and OBP:

•         OA = OB (radii of the same circle)

•         OP = OP (common side)

•         angle OAP = angle OBP = 90 degrees (radius perpendicular to tangent)

By RHS congruence: Triangle OAP is congruent to Triangle OBP

3.3 Tangent Length Formula

From the right triangle formed by the external point P, the centre O, and the point of tangency A:

where d = distance from external point to centre, r = radius of the circle.

3.4 Angle Between Two Tangents

When two tangents PA and PB are drawn from external point P, and angle AOB is the angle at the centre:

This arises because OAPB is a cyclic quadrilateral and opposite angles in a cyclic quadrilateral sum to 180 degrees.

4. Solved Examples

Example 1: Finding Tangent Length

Example 2: Angle Between Tangents

Example 3: Perimeter Using Equal Tangents

Example 4: Proving a Tangent

Example 5: Tangent to Incircle

5. Applications and Special Cases

5.1 Real-World Applications

•    Road and Railway Design: Tangent lines are used in the design of smooth curves where a straight section meets a curved section of a road or rail track.

•   Gear and Pulley Systems: The belt connecting two circular pulleys runs along the common external tangent, making tangent length calculations essential in mechanical engineering.

•   Architecture and Art: Circular arches and domes use chord and tangent geometry. Stained glass designs and tiling patterns rely on circle-tangent relationships.

•   Satellite Dish Alignment: The angle of a satellite dish is calculated using principles of tangent and secant lines from an external point.

•   Navigation: Ship pilots use the concept of tangent lines when calculating the closest point of approach to a circular obstacle such as a reef or island.

5.2 Special Case: Right-Angle Triangle from External Tangent

When a tangent is drawn from an external point P to a circle with centre O and point of contact A, the triangle OAP is always a right-angled triangle with the right angle at A. This means:

•   OP (hypotenuse) = sqrt(OA^2 + PA^2)

•   The angle at P and the angle at O are complementary (sum to 90 degrees)

•   The tangent length can always be found using Pythagoras Theorem

5.3 Special Case: Tangent-Chord Angle

The angle between a tangent drawn at a point on a circle and a chord drawn from the same point equals the inscribed angle subtended by the chord on the opposite side. This is called the Tangent-Chord Angle or the Alternate Segment Theorem and is used in higher geometry problems.

6. Formula Summary Table

All key formulas and results from this chapter are summarised below:

7. Key Theorems and Properties

7.1 Summary of NCERT Theorems

7.2 Important Corollaries

1.       The centre lies on the perpendicular bisector of the line segment joining the two points of contact from an external point.

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

3.       The line joining the external point to the centre also bisects the angle subtended by the chord of contact at the centre.

7.3 Properties of Tangent Lines

•         A tangent meets the circle at only one point and does not cross it.

•         Two circles can have 0, 1, 2, 3, or 4 common tangents depending on their relative positions.

•         Two circles that are externally tangent (touch externally) have 3 common tangents.

•         Two circles that are internally tangent (one inside the other, touching at one point) have 1 common tangent.

•         Two circles that do not intersect and one is outside the other have 4 common tangents.

7.4 Common Tangents to Two Circles

8. Common Mistakes and Exam Tips

8.1 Common Mistakes to Avoid

• Forgetting the perpendicularity condition: Students often fail to mark angle OAP = 90 degrees in diagrams. Always establish this before applying Pythagoras Theorem.

• Confusing equal tangent lengths with equal angles: PA = PB does not mean angle OPA = angle POA. Use congruent triangles properly.

• Missing the RHS step in theorem proof: In the proof of Theorem 10.2, the reason for RHS must be stated as radius perpendicular to tangent, not just right angle.

• Not using the quadrilateral angle sum: When asked for angle APB, students sometimes try to calculate directly instead of using angle APB + angle AOB = 180 degrees.

• Wrong variable assignment in perimeter problems: In circumscribed triangle/quadrilateral problems, assign tangent lengths from each vertex carefully using the equal tangent property.

8.2 Exam Tips for Full Marks

•  Always draw a neat, labeled diagram for every question. Mark the right angle at the point of contact, label the centre O, and show radii and tangent lines clearly.

•  In theorem proofs, write the statement, given, to prove, construction (if any), and proof separately. Each carries individual marks.

•  For tangent length problems, immediately set up a right triangle and apply Pythagoras. The formula PA = sqrt(OP^2 - r^2) should become instinctive.

•  In circumscribed polygon questions, always use the equal tangent property from each vertex to assign variables, then form an equation using the perimeter.

•  Learn to identify whether a line is a tangent or a secant from the diagram. If it touches at one point and the radius is perpendicular, it is a tangent.

9. Practice Questions

Test your understanding with the following questions arranged by marks:

Category A: 1 Mark Questions (MCQ / Very Short Answer)

• How many tangents can be drawn to a circle from a point inside it?

• The length of a tangent from a point 5 cm away from the centre of a circle of radius 3 cm is ___ cm.

• PA and PB are tangents from external point P. If PA = 7 cm, what is PB?

• The angle between a radius and the tangent at its outer endpoint is ___ degrees.

• Two circles of radii 5 cm and 3 cm touch externally. How many common tangents do they have?

• If angle APB = 50 degrees where PA and PB are tangents from P, find angle AOB.

Category B: 3 Mark Questions (Short Answer)

• Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

• Two tangents PA and PB are drawn to a circle with centre O from an external point P. Prove that angle OAB = angle OBA.

• A circle is inscribed in triangle ABC touching sides AB, BC, and CA at D, E, and F respectively. If AB = 14 cm, BC = 8 cm, and CA = 12 cm, find BD, CE, and AF.

• From an external point P, the tangent PA = 10 cm and the distance OP = 26 cm. Find the radius of the circle.

• Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Category C: 5 Mark Questions (Long Answer)

• Prove that the lengths of tangents drawn from an external point to a circle are equal. Use this to prove that if a circle is inscribed in a quadrilateral ABCD, then AB + CD = AD + BC.

• In a right triangle ABC, a circle is inscribed with radius r. If the legs are a and b and the hypotenuse is c, prove that r = (a + b - c) / 2.

• Two concentric circles of radii 5 cm and 3 cm are drawn. Find the length of the chord of the larger circle that is tangent to the smaller circle.

• In the given figure, XY and XY' are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersects XY at A and XY' at B. Prove that angle AOB = 90 degrees.

• A triangle ABC is circumscribed about a circle of radius 6 cm. If AB = 26 cm, BC = 30 cm, and CA = 28 cm, find the perimeter and verify the equal tangent property for each vertex.