NCERT Solutions Class 10 Maths Chapter 10 Exercise 10.1 Circles

NCERT Solutions for Class 10 Maths Chapter 10 – Circles Exercise 10.1 are provided to help students understand and solve problems related to properties of circles and theorems involving tangents, secants, and chords. These solutions are designed to simplify complex concepts and help students gain a clear understanding of circle-related geometry.

Exercise 10.1 focuses on:

NCERT Solutions Class 10 Maths Chapter 10 Exercise 10.1 Circles

Circles
Medium
Q.
How many tangents can a circle have?
Circles
Medium
Q.
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 _______.
Circles
Medium
Q.
A tangent PQ at a point P of a circle of radius 5 cm meets a line throughthe centre O at a point Q so that OQ = 12 cm. Length PQ isA 12 cm            B 13 cmC 8.5 cm           D 119 cm
Circles
Medium
Q.
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.
Circles
Medium
Q.
From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is
(A) 7 cm                                              (B) 12 cm
(C) 15 cm                                            (D) 24.5 cm

NCERT Solutions Class 10 Maths Chapter 10 Exercise 10.1 Circles

  • Basic properties of circles, such as tangents and secants.

  • Understanding and applying theorems related to the angle between a tangent and a chord.

  • Learning how to prove that the tangent at any point of a circle is perpendicular to the radius at the point of contact.

The solutions are presented in a step-by-step manner, helping students to grasp the concepts effectively and prepare confidently for their Class 10 exams.

Q1. How many tangents can a circle have?

Difficulty Level: Easy
Unknown:
Number of tangents a circle can have.

Reasoning:
A tangent to a circle is a line that intersects the circle at only one point. On every point on the circle, one tangent can be drawn.

Solution:
As per the above reasoning, a circle can have infinitely many tangents.

Q2. Fill in the blanks:

Difficulty Level: Easy
Solution:

(i) A tangent to a circle intersects it in _________ point(s).
Reasoning:
A tangent to a circle is a line that intersects the circle at only one point.

Answer:
One

(ii) A line intersecting a circle in two points is called a ____________.
Reasoning:
Secant is a line that intersects the circle in two points.

Answer:
Secant

(iii) A circle can have _________ parallel tangents at the most.
Reasoning:
Tangent at any point of a circle is perpendicular to the radius through the point of contact. Extended radius is a diameter which has two end points and hence two tangents which are parallel to themselves and perpendicular to the diameter.

Answer:
Two

(iv) The common point of a tangent to a circle and the circle is called _________.
Reasoning:
A tangent to a circle is a line that intersects the circle at only one point and that point is called as point of contact.

Answer:
Point of contact

Q3. A tangent PQ at a point P of a circle of radius 5 cm meets a line through the center O at a point Q so that OQ = 12 cm. Length PQ is:
(A) 12 cm (B) 13 cm (C) 8.5 cm (D) cm.

Difficulty Level: Easy
Known:
Radius OP = 5 cm
OQ = 12 cm

Unknown:
Length of the tangent PQ

Reasoning:
ΔOPQ is a right-angle triangle according to Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Solution:
By Pythagoras theorem,

OQ2=OP2+PQ2OQ^2 = OP^2 + PQ^2

122=52+PQ212^2 = 5^2 + PQ^2

144=25+PQ2144 = 25 + PQ^2

PQ2=144−25=119PQ^2 = 144 - 25 = 119

PQ=119≈10.91 cmPQ = \sqrt{119} \approx 10.91 \, \text{cm}

Answer:
Option D (approximately 11 cm)

Q4. 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.

Difficulty Level: Easy
Known:
(i) To draw a circle
(ii) Draw one tangent and one secant to the circle parallel to the given line.

Unknown:
To draw a circle as per known details.

Solution:
XY is the given line.
AB is the secant parallel to XY, AB // XY
AQ is the tangent parallel to XY, PQ // XY

FAQs: Class 10 Maths Chapter 10 – Circles Exercise 10.1

Q1. What is the main focus of Exercise 10.1?
Answer:
Exercise 10.1 focuses on the properties of circles, specifically on tangents, secants, and chords and applying theorems involving these properties to solve problems.

Q2. What is a tangent to a circle?
Answer:
A tangent to a circle is a straight line that touches the circle at exactly one point. This point is called the point of contact, and the tangent is perpendicular to the radius at this point.

Q3. What is the angle between the tangent and the radius at the point of contact?
Answer:
The angle between the tangent and the radius at the point of contact is always 90° (i.e., the tangent is perpendicular to the radius).

Q4. How do I apply theorems related to circles to solve problems?
Answer:
To solve problems, first identify the given information such as the radius, tangent, secant, or chord. Then, use relevant theorems like:

  • The tangent-secant theorem.

  • The tangent-chord theorem.

  • The angle between two tangents at a point.

Q5. How do NCERT Solutions help with exam preparation?
Answer:
These solutions provide clear and concise explanations for all the theorems and their applications, ensuring that students understand the properties of circles thoroughly and can solve related problems confidently in Class 10 exams.