Let a, b, c, d and p be any non-zero distinct | vedclass

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(C) The given equation is $(a^{2}+b^{2}+c^{2}) p^{2}-2(ab+bc+cd) p+(b^{2}+c^{2}+d^{2})=0$.This can be rewritten as $(a^{2}p^{2}-2abp+b^{2})+(b^{2}p^{2

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

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(C) The given equation is $(a^{2}+b^{2}+c^{2}) p^{2}-2(ab+bc+cd) p+(b^{2}+c^{2}+d^{2})=0$.This can be rewritten as $(a^{2}p^{2}-2abp+b^{2})+(b^{2}p^{2}-2bcp+c^{2})+(c^{2}p^{2}-2cdp+d^{2})=0$.This simplifies to $(ap-b)^{2}+(bp-c)^{2}+(cp-d)^{2}=0$.Since $a, b, c, d, p$ are real numbers,the sum of squares is zero only if each term is zero:$ap-b=0, bp-c=0, cp-d=0$.This implies $p = b/a = c/b = d/c$.Therefore,the ratio of consecutive terms is constant,which means $a, b, c, d$ are in $G.P.$

Let $a, b, c, d$ and $p$ be any non-zero distinct real numbers such that $(a^{2}+b^{2}+c^{2}) p^{2}-2(ab+bc+cd) p+(b^{2}+c^{2}+d^{2})=0$. Then:

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