If a, b, d and p are distinct non - zero real numbers such that `(a^2+b^2 + c^2) p^2 - 2(ab+bc+cd)p+(b^2 + c^2 +d^2) le 0` then n. Prove that a, b, c, d are in G. P and ad = bc

Explanation

`(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.