can u plz explain me how to find equation of common tangents (direct & traverse tangents )of a cirle in every cases...with an example..thanks

Working Rule to find direct common tangent:

Step I:      First find the point of intersection of direct common tangents say Q, which divides O₁ O₂ externally in r₁ : r₂

Step II:     Write the equation of any line passing through Q (α,β), i.e. y-β = m (x-α)…….(1)

Step III:    Find the two values of m, using the fact that the length of the perpendicular on (1) from the centre of one circle is equal to its radius.

Step IV:  Substitutes these values of ‘m’ in (1), the equation of the two direct common tangents can be obtained.

Working Rule to find transverse common tangent:

To fine the equations of transverse common tangent first find the point of intersection of transverse common tangents say P, which divides O₁O₂ internally in r₁:r₂. Then follow the step 2, 3 and 4.

Case II:    If the distance between the centres of the given circle is equal to sum of theirs radii. In this case both the circle will be touching each other externally. In this case two direct common tangents are real and distinct while the transverse tangents are coincident.

                        The point of contact P can be find by using the fact that it divides O₁ O₂ internally in r₁ :r₂ .

Case III:   It the distance between the centres of the given circles is equal to difference of their radii i.e. |O₁ O₂| = |r₁-r₂|, both the circles touches each other internally.

In this case point of contact divides O1 O2 externally in       r1 : r2.

In this case only one common tangent exist.

Case IV: It the distance between the centres of two given circle is less then the sum of their radii but greater then the difference of their radii i.e.

                  |r₁ - r₂| < O₁ O₂ < r₁ + r₂, in this case both the circle will intersect at two real and distinct points.