This page provides comprehensive Circles – Exercise 10.1. Get step-by-step NCERT solutions for Class 10 Maths Chapter 10 Circles Exercise 10.1. Understand Tangents and Secants.
Tangents to a Circle
Tangent: A line that intersects the circle at exactly one point.
Secant: A line that intersects the circle at two distinct points.
Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
1. How many tangents can a circle have?
A circle is made up of infinitely many points. Since a tangent can be drawn at each point of the circle, a circle can have infinitely many tangents.
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 _______.
(i) one (By definition of tangent)
(ii) secant
(iii) two (At the endpoints of a diameter)
(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) $\sqrt{119}$ cm
Given: Radius $OP = 5$ cm, $OQ = 12$ cm.
Since tangent is perpendicular to radius, $\angle OPQ = 90^\circ$.
In right $\triangle OPQ$, by Pythagoras theorem:
$OQ^2 = OP^2 + PQ^2$
$12^2 = 5^2 + PQ^2$
$144 = 25 + PQ^2$
$PQ^2 = 144 - 25 = 119$
$PQ = \sqrt{119}$ cm.
Option (D) is correct.
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.
1. Draw the circle with center O.
2. Draw the given line.
3. Draw a secant parallel to the given line (intersects at 2 points).
4. Draw a tangent parallel to the given line (touches at 1 point).