Question

If a,b and c are in G.P and the equations ax^2+2bx+c=0 and dx^2+2ex+f=0 have a common root,can you prove that d/a,e/b and f/c are in A.P?

Answer 1

Please see below.

Explanation:

Consider `ax^2+2bx+c=0`

As `a,b` and `c` are in G.P., let `b=ar` and `c=ar^2`

then the above becomes `ax^2+2arx+ar^2=0`

or `a(x^2+2rx+r^2)=0` i.e. `a(x+r)^2=0` and hence `x=-r` is the only root of `ax^2+2bx+c=0`.

i.e. `-r` is also the root of `dx^2+2ex+f=0`

and we have `dr^2-2er+f=0` and dividing this by `ar^2` we get

`d/a-(2e)/(ar)+f/(ar^2)=0`

or #`d/a-(2e)/b+f/c=0`

or `d/a-e/b=e/b-f/c`

Hence, `d/a,e/b` and `f/c` are in A.P.