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Number of Tangents from a Point on a Circle

Case 1
Let us take a point P inside a circle we find that all the lines through this point intersect the circle in 2 points. Hence we cannot draw any tangent to a circle through a point inside it.


Case 2
Now take a point P on the circle and draw tangents through this point. There can be only one tangent to the circle at such a point.

Case 3
If we now take a point P outside the circle and draw tangents to the circle from this point. We  find that exactly two tangents can be drawn to the circle through this point.


This can summarised as (case 1, case 2, case 3)

If T1 and T2 are the points of contact of the tangents PT1and PT2 respectively.

Here PT1 and  PT2 are the lengths of the tangents from the external point P. Also they are equal in length.  

Common Tangent 

A line which touches two given circles is called a common tangent to the circles.

There are two types of common tangent (i) Direct common tangent and (ii) Transverse common tangent.

Direct Common Tangent

If the common tangent lies on the same side of the centres of the circles, then it is called the direct common tangent.

Transverse Common Tangent

If the centres of the two circles lie on the opposite side of the common tangent, then it is called transverse common tangent.

Common tangents to different types of circles

(i) If two circles do not intersect, either two pairs of common tangents-- one pair of direct

common tangents and one pair of transverse tangents can be drawn or no common tangent can be drawn (when one lies completely inside the other). 

(ii) If two circles intersect at one point externally, one pair of direct common tangent and only one transverse common tangent can be drawn to the circles.

If two circles touch internally at one point, only one direct common tangent is possible.

(iii) If two circles intersect in two points, only one pair of direct common tangents can be drawn. 

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