www.free-education.in is a platform where you can get pdf notes from 6th to 12th class notes, General Knowledge post, Engineering post, Career Guidelines , English Speaking Trick , How to crack interview and lots more.Chapter 10 Circles

__Introduction to Circles__

__Introduction to Circles__

To know more about Circles, visit here.

Circle and line in a plane

For a circle and a line on a plane, there can be **three** possibilities.

i) they can be **non-intersecting**

ii) they can have **a single common point:** in this case, the line touches the circle.

ii) they can have **two common points:** in this case, the line cuts the circle.

(i) Non intersecting (ii) Touching (iii) Intersecting

**Tangent**

A **tangent to a circle **is a line which touches the circle at exactly one point. For every point on the circle, there is a unique tangent passing through it.

**Tangent**

To know more about Tangent, visit here.

**Secant**

A **secant to a circle** is a line which has two points in common with the circle. It cuts the circle at two points, forming a chord of the circle.

**Secant**

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Tangent as a special case of Secant

**Tangent as a special case of Secant**

The tangent to a circle can be seen as a special case of the secant when the two endpoints of its corresponding chord coincide.

**Two parallel tangents at most for a given secant**

For every given **secant** of a circle, there are **exactly two tangents which are parallel** to it and touches the circle at two **diametrically opposite points.**

**Parallel tangents**

Theorems

Tangent perpendicular to the radius at the point of contact

**Theorem**: The theorem states that “the **tangent **to the circle at any point is the **perpendicular to the radius** of the circle that passes through the point of contact”.

**Tangent and radius**

Here, O is the centre and OP⊥XY.

The number of tangents drawn from a given point

i) If the point is in an **interior region of the circle**, any line through that point will be a secant. So, **no tangent **can be drawn to a circle which passes through a point that lies inside it.

AB is a secant drawn through the point S

ii) When a point of tangency lies on the circle, there is **exactly one tangent** to a circle that passes through it.

**A tangent passing through a point lying on the circle**

iii) When the point lies outside of the circle, there are **accurately two tangents** to a circle through it

**Tangents to a circle from an external point**

**Length of a tangent**

The length of the tangent from the point (Say P) to the circle is defined as the segment of the tangent from the external point **P** to the point of tangency **I** with the circle. In this case, PI is the tangent length.

**Lengths of tangents drawn from an external point**

**Theorem: **Two tangents are of equal length when the tangent is drawn from an external point to a circle.

Tangents to a circle from an external point

PT1=PT2

To know more about Tangent Circle, visit here.