Circles.

Introduction to Circles

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

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

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 end points 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 radius at point of contact

Theorem : The tangent at any point of a circle is perpendicular to the radiusthrough the point of contact.

Tangent and radius

Here, O is the centre and $OP\perp XY$.

Number of tangents drawn from a given point

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

AB is a secant drawn
through the point S

ii) There is one and only one tangent to a circle passing through a point lying on the circle.

A tangent passing through a point
lying on the circle

iii) There are exactly two tangents to a circle through a point lying outside the circle.

Tangents to a circle from an
external point

Length of a tangent

The length of the segment of the tangent from the external point P to the point of contact I with the circle is called the length of the tangent from the point P to the circle.

PI is the length of a tangent

Lengths of tangents drawn from external point

Theorem : The lengths of tangents drawn from an external point to a circle are equal.

Tangents to a circle from an
external point

$P{T}_1=P{T}_2$