### CHAPTER AT A GLANCE

**1. Introduction**

A circle is a set of all points in a plane at a fixed distance from a fixed point in a plane. The fixed point is called the centre of the circle. The fixed distance is called the radius of the circle.

**2. Line and a Circle**

In Fig. (i), the line PQ and the circle have no common point. In this case, PQ is called a non-intersecting line with respect to the circle. In Fig. (ii), there are two common points A and B that the line PQ and the circle have. In this case, we call the line PQ a secant of the circle. In Fig. (iii), there is only one point A which is common to the line PQ and the circle. In this case, the line is called a tangent to the circle.

**3. Tangent**A tangent to a circle is a straight line that touches the circle at only one point. The point where the tangent touches the circle is called point of con tact of the tangent to the circle.

A tangent to a circle is a special case of a secant when the two end points of its corresponding chord coincide.

**(I) Theorem: **Tangent at any point on a circle is perpendicular to the radius through the point of contact.

CB is the tangent to the given circle touching at A and OA is the radius.

:. LOAB=90°**(I)** At any point on the circle there can be one and only one tangent.**(ii)** The line containing the radius through the point of contact is called the normal to the circle at the point.

**4. Number of Tangents from a Point to Circle****(I)** No tangent can be drawn from the point lying inside the circle, as shown in fig.(i).**(ii)** One and only one tangent can be drawn from a point lying on the circle, as shown in fig.(ii).**(iii)** Only two tangents can be drawn from an exterior point to a circle, as shown in fig.(iii).

**5. Length of a Tangent**

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

In the given figure, T1 & T2 are the points of contact of the tangents PT1 & PT2 respectively from the external point P.

**6. Theorem Related to Length of tangents**

**(I) From the External Points**

The lengths of tangents drawn from an external point to a circle are equal. i.e.,

Here, PQ & PR are the two tangents drawn from P to the circle.

Here OP is the angle bisector of LQPR, i.e., the centre lies on the bisector of the angle between the two tangents

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