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

[Sumedh]

Homework Statement

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.

Homework Equations

The Attempt at a Solution

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?

 

[PeterO]

Homework Statement

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.

Homework Equations

The Attempt at a Solution

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?

You realize that fro an AP, where each term is obtained by adding a fixed amount, you add the 1st and 3rd and divide by 2 to get the 2nd or middle.
In a GP where terms are obtained by multiplying by a fixed amount, you can multiply the 1st an 3rd then take the second root to get the 2nd or middle term. [second root = square root]

 

[Sumedh]

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