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If a ,b ,c ,da n dp are distinct real nu...

If `a ,b ,c ,da n dp` are distinct real numbers such that `(a^2+b^2+c^2)p^2-2(a b+b c+c d)p+(b^2+c^2+d^2)lt=0,` then prove that `a ,b ,c , d` are in G.P.

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`(a^(2)+b^(2)+c^(2))p^(2)-2(ab+bc+cd)p+(b^(2)+c^(2)+d^(2))ge0`
`rArr(ap-b)^(2)++(bp-c)^(2)+(cp-d)^(2)ge0` (1)
Since a,b,c,d and p are real, the inequality (1) is possible only when each of the factors is zero, i.e.,
ap-b=0,bp-c=0,cp-d=0
`rArrp=b/a=c/b=d/c`
`rArr` a,b,c,d are in G.P.
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