Maths

CIRCLES 209
Take a point Q on XY other than P and join OQ (see Fig. 10.5).
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.)
Fig. 10.5
Remarks :
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.
1. How many tangents can a circle have?
2. Fill in the blanks :
EXERCISE 10.1
(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) 12 cm (B) 13 cm (C) 8.5 cm (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.
10.3 Number of Tangents from a Point on a Circle
To get an idea of the number of tangents from a point on a circle, let us perform the
following activity: