In Fig. 10.1 (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. 10.1 (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. 10.1 (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.
CIRCLES
Fig 10.1 (i)
Fig 10.1 (ii)
Fig 10.1 (iii)
Theorem 10.1 : The tangent at any point of a circle is perpendicular to the radius through the point of contact.
O
X Q P Y
Take a point Q on XY other than P and join OQ (see Fig).
The point Q must lie outside the circle. (Why? Note that if Q lies inside the circle, XY will become a secant and not a tangent to the circle). Therefore, OQ is longer than the radius OP of the circle. That is, OQ > OP.
Since this happens for every point on the line XY except the point P, OP is the shortest of all the distances of the point O to the points of XY. So OP is perpendicular to XY. (as shown in Theorem A1.7.)
1. By theorem above, we can also conclude that at any point on a circle there can be one and only one tangent.
2. The line containing the radius through the point of contact is also sometimes called the ‘normal’ to the circle at the point.
EXERCISE
1. How many tangents can a circle have?
2. Fill in the blanks :
(i) A tangent to a circle intersects it in __________ point (s).
(ii) A line intersecting a circle in two points is called a __________.
(iii) A circle can have ____________ parallel tangents at the most.
(iv) The common point of a tangent to a circle and the circle is called ___________.
.
3. A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q
so that OQ = 12 cm. Length PQ is :
(A) 12cm (B) 13cm (C) 8.5cm (D) $√{119}$ cm.
4. Draw a circle and two lines parallel to a given line such that one is a tangent and the other,
a secant to the circle.