Let a , b, c , d and p be any non zero distinct real numbers such that (a2 + b2 + c2)p2 --…

Question

Let a , b, c , d and p be any non zero distinct real numbers such that

(a2 + b2 + c2)p2 -- 2(ab + bc + cd)p + (b2 + c2 + d2) = 0. Then :

Accepted Answer

(a2 + b2 + c2)p2 -- 2(ab + bc + cd)p + (b2 + c2 + d2) = 0

$$ \Rightarrow $$ (a2p2
+ 2abp + b2
) + (b2p2
 + 2bcp + c2
) + (c2
p2
+ 2cdp + d2) = 0

$$ \Rightarrow $$ (ab + b)2
 + (bp + c)2
 + (cp + d)2
 = 0

Note : If sum of two or more positive quantity is zero then they are all zero.

$$ \therefore $$ ap + b = 0 and bp + c = 0 and cp + d = 0

p = $$ - {b \over a}$$ = $$ - {c \over b}$$ = $$ - {d \over c}$$

or $${b \over a}$$ = $${c \over b}$$ = $${d \over c}$$

$$ \therefore $$ a, b, c, d are in G.P.

Solutions · 1