  Circles
All
Guides
Practice
Learn

# NCERT Solutions for Class 10 Maths Chapter 10 : Circles

Circles Class 10 Maths NCERT Solutions Chapter 10 is concerned with the understanding of various aspects of circles. Circles are essential in mathematics because they are covered extensively in the geometry curriculum. Students preparing for their Class 10 exams will be able to clear all their concepts on circles at the root level. Expert faculty of Toppr produced these solutions to assist students with their first term exam preparations. It covers all major concepts in detail, allowing students to understand the ideas better.

Circles Class 10 NCERT Solutions Circles, the tenth chapter of the section, concentrates on the essential concepts such as the tangents, the lengths of tangents, normal to a circle, the radius-tangent relationship, and so on. The properties based on the above mentioned topics are used to solve questions related to these topics step by step. All of these solutions are designed with the new CBSE pattern in mind so that students have a complete understanding of their tests.

Circle Class 10 Chapter 10 Questions and Answers are very useful for getting good grades in tests and properly preparing you with all of the important concepts. These NCERT Solutions are valuable tools that can assist you not only in covering the full syllabus but also in providing an in-depth analysis of the subjects. The Chapter 10 Maths Class 10 NCERT Solutions are available in pdf format below, and some of them are also included in the exercises.

Table of Content
Learn with Videos

## Access NCERT Solutions for Class 10 Maths Chapter 10 : Circles

Exercise 10.1
Question 1
How many tangents can be drawn at any point on  the circle?
A
B
C
D
Medium
Solution Verified by Toppr

Let P be any point on the circle with center

Take a line L through P and Q as shown if L is perpendicular to OP

because the perpendicular distance is shortest.

Every point except P lies outside the circle and line must be a tangent. At any given point one and only one tangent can be drawn.  Question 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) A common point of a tangent to a circle and the circle is called ______.
Easy
Solution Verified by Toppr
(i) One point
Circle is the locus of points equidistant from a given point, the center of the circle, and nd Tangent is the line which intersect circle at one point only

(ii) Secant
A line which intersect circle in two points is called a chord if this line passes through center then it is called secant. (Points are A and B)

(iii) Two
As circle has infinite points, so there will be infinite tangents can be drawn on these points which touches at only one point. (P and Q)

So there will be infinite pairs of tangents which are parallel.

(iv) Point of Contact
The common point of a tangent to a circle and the circle is called point of contact. (P or Q in the given figure)  Question 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 cm. Length of PQ is :
A
cm
B
cm
C
cm
D
cm
Medium
Solution Verified by Toppr
Given,
PQ is the tangent to the circle.
So,by Pythagoras theorem we get,
cm

Question 4
Draw a circle and two lines parallel  to a given line that one is a tangent and the other, a secant to the circle
Medium
Solution Verified by Toppr
Let the circle has a center and be a given line such that
Let .
Where is a second ans is a tangent intersecting the circle at .  Exercise: 10.2
Question 1
From a point , the length of the tangent to a circle is cm and the distance of from the centre is cm. The radius of the circle is
Medium
Solution Verified by Toppr
Tangent=

point of contact=
length of the tangent to a circle =
i.e.,
prove
Proof
Since is tangent
so,
So,
is a right angled triangle using by pythagoras
theorem
Hence ,
Question 2
If and are two tangents to a circle with centre so that , then is equal to
A
B
C
D
Medium
Solution Verified by Toppr
Given,
We know,
(Angle between the tangent and the radial line at the point of intersection of the tangent at the circle)
Sum of angles  Question 3
If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of , then  is equal to:
Medium
Solution Verified by Toppr

Given that,

PA and PB are two tangents a circle and

To find that

Construction:- join

Proof:-  Since       and

Then         and

In

So,

So,

In

The value of is  Question 4
Prove that the tangents drawn at the end of a diameter of a circle are parallel
Medium
Solution Verified by Toppr
To prove: PQ RS
Given: A circle with centre O and diameter AB. Let PQ be the tangent at point A & Rs be the point B.
Proof: Since PQ is a tangent at point A.
OA PQ(Tangent at any point of circle is perpendicular to the radius through point of contact).
…………
OB RS
……………
From &
i.e.,
for lines PQ & RS and transversal AB
i.e., both alternate angles are equal.
So, lines are parallel.
\therefore PQ||RS.  Question 5
Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre of the circle.
Medium
Solution Verified by Toppr
Given a circle with center and the tangent intersecting circle at point
and prove that
We know that tangent of the circle is perpendicular to radius at points of contact Hence
So,
Now lets assume some point
Such that
Hence
From eq &
Which is possible only if line passes though
Hence perpendicular to tangent passes though centre  Question 6
The length of a tangent from a point A at distance cm from the centre of circle is cm. The radius of the circle is
A
cm
B
cm
C
cm
D
cm
Medium
Solution Verified by Toppr
Let be the tangent drawn from a point to a circle with centre

and cm and cm. Since tangent at a point is

perpendicular to the radius through the point of contact

from right angled ,

cm  View more
Practice more questions

### NCERT Solutions for Class 10 Maths Chapter 10 Circles – Brief Overview

#### 1. Circles

A circle is a two-dimensional shape formed by all points in a plane that are equidistant from a specific point, called the center.

#### 2. Tangent to a Circle

The tangent line to a plane curve at a given location in geometry is the straight line that "almost touches" the curve at that point. When the two endpoints of a chord coincide, the tangent to a circle is a specific case of the secant. A tangent to a circle is a line that only touches the circle once.

Few properties of tangents to a circle:

• A circle's tangent is perpendicular to the radius via the point of contact.
• The lengths of the two tangents to a circle from an exterior point are equal.

#### 3. Number of Tangents From a Point

• A tangent to a circle from within the circle does not exist.
• A single tangent to a circle exists from a single point on the circle.
• There are two possible tangents to a circle from a point outside the circle.
Related Chapters

### Frequently Asked Questions on NCERT Class 10 Maths Chapter 10 : Circles

Q1. What are the most important topics in NCERT Class 10 Chapter 10 Circles?

Answer: NCERT Solutions for Class 10 Maths Chapter 10 covers the following topics: introduction to circles, tangent to a circle, lengths of tangents, number of tangents from a point on a circle, the radius-tangent relationship, and the chapter summary.

Q2. How many exercises are there in Chapter 10 of Class 10 Maths?

Answer: NCERT Solutions Class 10 Maths Chapter 10 Circles contains a total of 17 questions separated into 2 exercises. Class 10 Maths Chapter 10 Circles contains a total of 17 sums, 10 of which are simple, 3 of which are fairly complicated, and 4 of which are tough.

Q3. How many tangents can a circle have?

Answer: Generally, there is no limit to the number of tangents a circle can have. It can be limitless since a circle is made up of infinite points that are all equally far from one another. Because there are infinite locations on the circumference of a circle, infinite tangents can be traced from them.

Q4. Do I have to practice all of the questions in NCERT Solutions Class 10 Maths Chapter 10 Circles?

Answer: All questions in NCERT Solutions Class 10 Maths Circles are based on key concepts that provide a comprehensive overview of the subject. Because the problems can be complex, practising all of the sums, as well as the examples, can help students learn the idea of circles and understand how to use various formulas. Students must also revise the given theorems and practice questions twice because the sums are proof-based.

Related Chapters