quadratic equation, A.P. and G.P. related problem problem

if ax2+2bx+c=0 and a1x2+2b1x+c1 have a common root and

a/a1 ,b/b1 ,c/c1 are in A.P.

show that a1,b1,c1 are in G.P.

I know the mean formula of A.P. i.e. the middle term is the mean of the other two.

any hints of which formula of G.P. to use and how to solve???

Re: quadratic equation, A.P. and G.P. related problem problem

What does A.P and G.P mean? (sorry I'm not from here)

Re: quadratic equation, A.P. and G.P. related problem problem

A.P. means arithmetic progression

G.P. means geometric progression

Re: quadratic equation, A.P. and G.P. related problem problem

thank you i got the answer.

let the AP be

(A-D) , (A), (A+D)

then

a=a1(A-D)

b=b1(A)

c=c1(A+D)

one root is common so

putting these values in the formula

(c1a2-c2a1)^2 = (a1b2-a2b1)(b1c2-b2c1)

we get the required proof.

thank you very much