Circles Class 10 CBSE Maths Chapter 10

Class 10 CBSE Maths Syllabus
Chapter 1: Real Numbers
Chapter 2: Polynomials
Chapter 3: Linear Equations in Two Variables
Chapter 4: Quadratic Equations
Chapter 5: Arithmetic Progression
Chapter 6: Triangles
Chapter 7: Coordinate Geometry
Chapter 8: Introduction to Trigonometry
Chapter 9: Applications of Trigonometry
Chapter 10: Circles
Chapter 11: Constructions
Chapter 12:Areas Related to Circles
Chapter 13: Surface Areas and Volumes
Chapter 14: Stastics
Chapter 15: Probability

Introduction

Below we have provided important points to remember for NCERT Class 10 Maths Chapter 10:

A tangent to a circle is a line that intersects the circle at only one point.
Tangent to a circle at a point is perpendicular to the radius through the point of contact.
The tangent to a circle is a special case of the secant, when the two end points of its corresponding chord coincide.
From a point on the circle only one tangent can be drawn.
From a point, lying outside a circle, two and only two tangents can be drawn to it.
A circle is a collection of all points in a plane which are at a constant distance (radius) from a fixed point called its centre. Class 10th students are already familiar with the fundamental concepts of circles like the chord, segment, sector, arc etc. The Exercise 10.1 of Circles is based on the topics like non-intersecting lines, secants and tangents to circles.
If we consider a circle and a line PQ, then,
(i) If the line PQ and the circle have no common point, the line PQ is called a non-intersecting line with respect to the circle.
(ii) If there are two common points, say, A and B, that the line PQ and the circle have, then we call the line PQ a secant of the circle.
(iii) If there is only one point, say A, which is common to the line PQ and the circle, then the line PQ is called a tangent to the circle.
Tangent to a Circle: A tangent to a circle is a line that intersects the circle at only one point. The tangent to a circle is a special case of the secant when the two endpoints of its corresponding chord coincide. The common point of the tangent and the circle is called the point of contact, and the tangent is said to touch the circle at the common point.
Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Proof: We are given a circle with centre O and a tangent XY to the circle at a point P. We need to prove that OP is perpendicular to XY.
Take a point Q on XY other than P and join OQ (see the figure at the bottom of this section).
The point Q must lie outside the circle. If Q lies inside the circle, XY will become a secant and not a tangent to the circle. Therefore, OQ is longer than the radius OP of the circle. That is,
OQ > OP.
Since this happens for every point on the line XY except the point P, OP is the shortest of all the distances from point O to points of XY. So OP is perpendicular to XY.