Considerring, that my assumption of the question is correct,

 

you can try solving it like this:-

 

Let the common root be m

The equations, will then hold true for x=m.

now put x=m

am²+bm+c=0

and a1m²+b1m +c1=0

Now, try finding the relation between a1, b1,c1 and a,b,c.

You will get another relation from the fact that a/a1, b/b1, c/c1 are in A.P.

Using these two relations , you will get the desired relation of a1, b1, c1.

O.K.?????????

 

rate me if I have been of any help to you.