CHAPTER AT A GLANCE
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.
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.
(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