How to find number of common tangents between two circles ?

         Finding    common   tangents    b/w    two    circles

Q.). What is the meaning of  common tangents ?

Ans.)  Its  meaning is as simple as its name, a common tangent is a tangent that is tangent to both the given curves. These curves may be two parabolas , one parabola and a circle or may be hyperbola and a parabola. Actually, there are immense possibility what these curves can be. But we will specifically discuss about two circles in this post.

There are maximum " 4 " possible common tangents b/w to circles and minimum is zero. OK lets understand by diagrams.

Now, we will discuss how to calculate number of common tangents b/w two circles :

Let C1 and C2 be two circles with r1 and r2 be there respective radius and (x1, y1) and (x2, y2) be the centre of C1 and C2 respectively.

Let " D " be the distance b/w centre of two circles C1 and C2.

Now, If

1)              r1 + r2   <   D               (  4  common tangents  )

2)            r1  +   r2  =   D               (  3 common tangents ) 

3)             r1   +   r2   >  D   AND     | r1   -    r2 |    <   D   ( 2 common tangents )

4)             r1   +   r2   >  D   AND     | r1   -    r2 |    =   D   (  1 common tangent )

5)             r1   +   r2   >  D   AND     | r1   -    r2 |    >    D   ( No common tangent ) 

Use above figure to have a better understanding of these cases. 

Let's also take an example :

C1  :   r1 =   5 with centre (0 , 0)

C2  :  r2 = 3 with centre ( 0 ,  7)

Now,   

r1 + r2 = 5 + 3 = 8

D = 7 ( distance b/w centres )

As, we can clearly see it is  " case 3 "  so we have two common tangents .