The number of common tangents for concentric circles is zero.

The number of common tangents for two circles touching internally is one.

The number of common tangents for two intersecting circles is  two.

The number of common tangents for two circles touching externally is three out of which two are direct tangent and the third is an indirect tangent.

The number of common tangents for two non intersecting and not touching circles is four. out of which two are direct and two are inditrect.

Solution : Let the common tangent at P touches the common tangent AB at T

        AT = PT         (1)   [ Length of tangent from point T are equal]

        BT= PT          (2)   [ Length of tangent from point T are equal]

From (1) and (2) We get     AT = BT

Hence T bisects AB

In APT                        (3)   [ angle opposite equal sides are equal]

In BPT                       (4)    [ angle opposite equal sides are equal]

  adding (3) and (4)

        (5)

  In

         [ Angle sum property]

                         [Using Eq (5) ]

Hence AB subtends angle of  right angle at P