NCERT Solutions Class 10 Maths Chapter 10 Circles Exercise 10.1

NCERT Solutions Class 10 Maths Chapter 10 Exercise 10.1 will help you extend your knowledge of circles. The exercise is focused on the properties of tangents to circles and solving questions related to these tangents. The topic includes many foundational concepts of circles in geometry and their relationship with lines. The exercise forms a solid base for more complex topics that students may face in higher classes. So, let’s start exploring this important topic not only for exams but also for future mathematical studies.

1.0Download NCERT Solutions Class 10 Maths Chapter 10 Circles Exercise 10.1 : Free PDF

2.0Introduction to Circles and Lines

A circle is a two-dimensional geometrical figure, which is also referred to as the collection of all points in a plane at a constant distance from a single point, known as the centre. At the same time, a line is a one-dimensional figure which extends in both directions without ending. In general, a line is represented as a straight path with arrows at both ends, which indicates its never-ending property. In the exercise, we will learn the relation between the circles and lines, which ultimately helps in solving questions asked in this section of the chapter. 

3.0Class 10 Maths Chapter 10 Circles Exercise 10.1: Key Concepts

Lines and Their Interactions with a Circle:

Lines interact with a circle in different ways. To understand these interactions, consider a line AB and a circle in a single plane. Here, AB will form three possible scenarios with a circle: 

• Non-intersecting Line: In this case, the line does not cut the circle at all. This kind of line is referred to as a non-intersecting line.
• Secant: If the line cuts the circle at two points, P and Q, then the line is referred to as a secant.
• Tangent: If the line intersects the circle at a single point, P, then it is referred to as a tangent.

Among the above-mentioned concepts, exercise 10.1 deeply explores the concepts of tangents. Let’s take a deeper dive into these concepts. 

Tangent to a Circle:

The word tangent is associated with the Latin word “Tangere” which simply means to touch, a classical property of a tangent. A tangent can also be described as the special case of a secant, where the two ends of its corresponding chord coincide. The point at which a tangent touches the circle is known as the point of contact of a tangent. 

In real life, the point of contact can be understood by an example of a wheel. The point where the wheel touches the ground is the point of contact, and then the ground is considered the tangent of the wheel.   

Tangent and Radius of Circle: 

In the exercise, the most important property of a tangent is its relation with the radius of the circle. According to this, a tangent to a circle at any point is always perpendicular to the radius at the point of contact. This statement is formulated as a theorem, which is proved using the method of contradiction. 

Remarks: 

• As per the above theorem, we can say that at the point of contact on a circle, there can be only one tangent, meaning a tangent is unique to its point of contact. 
• As the radius is always perpendicular to the point of contact of a tangent, it is sometimes referred to as the normal to the circle. 

These concepts discussed earlier are important to form a basic understanding of circles. This is why, you should start practising and master the concepts of tangents, secants, and circles at ease by using NCERT Solutions Class 10 Maths Chapter 10 Exercise 10.1 – your complete guide to excelling geometry. 

4.0NCERT Class 10 Maths Chapter 10 Circles Exercise 10.1 : Detailed Solutions

1. How many tangents can a circle have?

Sol. There can be infinitely many tangents to a circle.

2. Fill in the blanks :

(i) A tangent to a circle intersects it in ..... point (s).

(ii) A line intersecting a circle in two points is called a.......

(iii) A circle can have ……… parallel tangents at the most.

(iv) The common point of a tangent to a circle and the circle is called …….

Solution:

(i) One

(ii) Secant

(iii) Two

(iv) Point of contact.

3. A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length PQ is.

(A) 12 cm

(B) 13 cm

(C) 8.5 cm

(D) √119 cm

Solution:

0 is the centre of the circle. The radius of the circle is 5 cm.

PQ is tangent to the circle at P. Then OP = 5 cm and ∠OPQ = 90°.

We are given that OQ = 12 cm.

By Pythagoras Theorem, we have:

PQ² = OQ² - OP²

= (12)² - (5)² = 144 - 25 = 119

⇒ PQ = √119 cm

Hence, the correct option is (D).

4. Draw a circle and two lines parallel to a given line such that one is a tangent and the other a secant to the circle.

Solution:

We have the required figure as shown:

Here, l is the given line and a circle with center O is drawn.

The line n is drawn which is parallel to l and tangent to the circle. Also, m is drawn parallel to line l and is a secant to the circle.

5.0Benefits of Studying NCERT Solutions Class 10 Maths Chapter 10 Exercise 10.1

• Helps in understanding the fundamentals of circles and tangents.
• Provides clear, structured solutions to improve problem-solving skills.
• Strengthens your grasp of the topic, making students more confident.
• Helps in practicing efficiently, improving speed and accuracy.
• Builds a strong base for advanced math in higher classes.