# NCERT Class 10 Maths Circles

The chapter 10 on Circles begins with the introduction of the relationship with a line and a circle. Then the concept of tangents to a circle and a theorem on the relationship between a tangent and the radius of the circle is presented. Different cases of the number of tangents that can be drawn to a circle and the related theorem is discussed in the later sections of the chapter. Important results regarding the length of the tangents drawn to a circle are also discussed in this chapter. The problems are mainly based on the theorems to determine the properties and length of the tangent drawn to the circle.

Download FREE PDF of Chapter-10 Circles

Exercise 10.1

## Chapter 10 Ex.10.1 Question 1

How many tangents can a circle have?

**Solution**

**Video Solution**

**What is 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.

**Steps:**

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

## Chapter 10 Ex.10.1 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) The common point of a tangent to a circle and the circle is called _________ .

**Solution**

**Video Solution**

**Steps:**

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

**Reasoning:**

A tangent to a circle is a line that intersects the circle at only one point.

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

**Reasoning:**

Secant is a line that intersects the circle in two points.

(iii) A circle can have ** Two **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.

Center \(O,\) diameter \(AB,\text{ tangents}\, PQ, RS\) and, \({PQ}\; \| \;{RS}\)

\(A\) and \(B\) are called as point of contact.

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

**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.

## Chapter 10 Ex.10.1 Question 3

A tangent \(PQ\) at a point \(P\) of a circle of radius \(\text{5 cm}\) meets a line through the center \(O\) at a point \(Q\) so that \(OQ = \text{12 cm.}\) Length \(PQ\) is:

(A) \(\text{12 cm}\)

(B) \(\text{13 cm}\)

(C) \(\text{8.5 cm}\)

(D) \(\sqrt{119} \,\rm{cm.}\)

**Solution**

**Video Solution**

**What is Known?**

Radius \(OP = \text{5 cm} \)

\(OQ = \text{12 cm}\)

**What is Unknown?**

Length of the tangent \(PQ\)

**Reasoning:**

\(\Delta {OPQ}\) 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.

**Steps:**

By Pythagoras theorem

\[\begin{align} {OQ} ^ { 2 } & = {O P} ^ { 2 } + {P Q} ^ { 2 } \\ 12 ^ { 2 } & = 5 ^ { 2 } + {P Q} ^ { 2 } \\ 144 & = 25 + {P Q} ^ { 2 } \\ {P Q} ^ { 2 } & = 119 \\ {P Q} & = \sqrt { 119 } \rm {\;cm } \end{align}\]

The answer is option D.

## Chapter 10 Ex.10.1 Question 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**

**Video Solution**

**What is Known?**

(i) To draw a circle

(ii) Draw one tangent and one secant to the circle parallel to the given line.

**What is Unknown?**

To draw a circle as per known details.

**Steps:**

\(XY\) is the given line.

\(AB\) is the secant parallel to \(XY,\) \({AB}\parallel {XY}\)

\(AQ\) is the tangent parallel to \(XY,\) \({PQ}\parallel {XY}\)