Question

Let a,b,cd be real numbers in G.P If u,v,w satisfy the system of equations
$u+2v+3w=6$\\
$4u+5v+6w=12$\\
$6u+9v=4$\\
Then show that the roots of the equation \\
$[\frac{1}{u} +\frac{1}{v} + \frac{1}{w}]x^2 + [(b-c)^2+(c-a)^2+(d-b)^2]x + u+v+w=0$
and $20x^2+10(a-d)^2x-9=0$ are reciprocals of each other.

Answer 1

on solving,,we get u= -1/3,v= 2/3 ,w = 5/3
from 1st eqn,,product of roots = -20/9..let roots be p,q => pq = -20/9
and from 2nd eqn,,product = -9/20..let roots be r,s => rs = 9/20
now (b-d)^2 + (b-c)^2 + (c-a)^2= 2b^2 + 2c^2 + d^2 + a^2 - 2ac-2bd-2bc
now bd=c^2 and ac=b^2
so we have d^2 - 2bc +a^2
bc=ad
=>d^2 - 2ad + a^2 = (d-a)^2

so,,easier now i guess?

Answer 2

thanks a lot once again....

Answer 3

you're welcome ONCE AGAIN! :D