# 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.

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**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. WeÂ find that exactly two tangents can be drawn to the circle through this point.

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

If T_{1} and T_{2} are the points of contact of the tangents PT_{1}and PT_{2} respectively.

Here PT_{1 }andÂ PT_{2} are the lengths of the tangents from the external point P. Also they are equal in length. Â

# Common TangentÂ

A line which touchesÂ 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).Â

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**(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.

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

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

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