**Circles**

Contents

### C__ircles__

A circle is a collection of all points in a plane which are at a constant distance (radius) from a fixed point (centre). We will study Secant & Tangent

Circles are used extensively in our day to day life. Car tyres, football, well, CD, DVDs etc is circular.

Circles are used extensively in our day to day life. Car tyres, football, well, CD, DVDs etc is circular.

__Circle & a Line__

If we consider a circle and a line, there can be three possibilities.

- A non-intersecting line AB with respect to the circle. In this case there is no common point between line AB & circle
- Line CD intersecting the circle at
**one**point G. There is only one point G which is common to the line and the circle. In this case, the line CD is called a tangent to the circle. - Line EF intersecting the circle at two points H & I. There are two common points H and I that the line EF and the circle have. In this case, we call the line EF a secant of the circle.

__Tangent to a Circle__

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

There is only one tangent at a point of the circle.

A circle can have two parallel tangents at the most.

**Theorem 1:**The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Refer ExamFear video lessons for Proofs.

The line containing the radius through the point of contact is also sometimes called the ‘normal’ to the circle at the point.

__Number of tangents from a point on a Circle__

If we consider 3 scenarios for the point

- Point inside circle : 0 tangents
- Point on the circle : 1 tangent
- Point outside the circle: 2 tangents

The length of the segment of the tangent from the external point P and the point of contact with the circle is called the

**length of the tangent**from the point P to the circle. PT1 and PT2 are the lengths of the tangents from P to the circle. The lengths of tangents PT1 and PT2 are equal.**Theorem 2 :**The lengths of tangents drawn from an external point to a circle are equal.

Refer ExamFear video lessons for Proofs.

**Numerical**: Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

Solution: We have to find length of AC.

∠OBC = 90o, since AC is chord to smaller cirle.

In right triangle OBC, BC2 =OC2 –OB2= 52 -32

Thus BC = 4 cm

We can also prove that Δ OBC ≅ Δ OBA, thus AB = BC

So AB = 2* BC = 2* 4 cm = 8 cm.

**Also Read :**

### MATHS Revision Notes

Chapter:01 Real Numbers System

Chapter:02 Polynomials

Chapter:03 Pair of Linear Equations in Two Variables

Chapter:04 Quadratic Equation

Chapter:05 Arithemetic Progressions

Chapter:06 Triangles

Chapter:07 Coordinate Geometry

Chapter:08 Introduction to Trignometry

Chapter:09 Some Application Of Trignometry

Chapter:10 Circles

Chapter:11 Constructions

Chapter:12 Area Related to Cirles

Chapter:13 Surface Area Volume

Chapter:14 Stastistics

Chapter:15 Probability