NCERT Solutions Class 10 Maths Chapter 10: Circles
Detailed solutions for Exercise 10.1 (Introduction to Tangents).
#### **Q1. How many tangents can a circle have?**
#### **A1. Solution:**
A circle is made up of infinitely many points. Since a tangent can be drawn at every single point on the circle's boundary, a circle can have **infinitely many** tangents.
#### **Q2. 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 __________.**
#### **A2. Solution:**
(i) **exactly one**
(ii) **secant**
(iii) **two**
(iv) **point of contact**
#### **Q3. 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) $\sqrt{119}$ cm**
#### **A3. Solution:**
The radius is perpendicular to the tangent at the point of contact.
Therefore, $\angle OPQ = 90^\circ$.
In right-angled $\Delta OPQ$, by Pythagoras theorem:
$OQ^2 = OP^2 + PQ^2$
$(12)^2 = (5)^2 + PQ^2$
$144 = 25 + PQ^2$
$PQ^2 = 144 - 25$
$PQ^2 = 119$
$PQ = \sqrt{119}$ cm.
**Correct Option: (D)**
#### **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.**
#### **A4. Solution:**
1. Draw a circle with any centre $O$.
2. Draw a given line $l$ (outside the circle).
3. Draw a line $m$ parallel to line $l$ such that it touches the circle at exactly one point. Line $m$ is the **tangent**.
4. Draw another line $n$ parallel to line $l$ such that it intersects the circle at two distinct points. Line $n$ is the **secant**.