If a, b, c, d and p are distinct real numbers | vedclass

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(D) The given inequality is $(a^2 + b^2 + c^2)p^2 - 2p(ab + bc + cd) + (b^2 + c^2 + d^2) \le 0$.This can be rearranged as:$(a^2 p^2 - 2abp + b^2) + (b

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

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(D) The given inequality is $(a^2 + b^2 + c^2)p^2 - 2p(ab + bc + cd) + (b^2 + c^2 + d^2) \le 0$.This can be rearranged as:$(a^2 p^2 - 2abp + b^2) + (b^2 p^2 - 2bpc + c^2) + (c^2 p^2 - 2cpd + d^2) \le 0$.This simplifies to:$(ap - b)^2 + (bp - c)^2 + (cp - d)^2 \le 0$.Since the sum of squares of real numbers can only be non-negative,the only way for the sum to be less than or equal to $0$ is if each term is exactly $0$.Therefore,$ap - b = 0$,$bp - c = 0$,and $cp - d = 0$.This implies $p = \frac{b}{a} = \frac{c}{b} = \frac{d}{c}$.This is the definition of a Geometric Progression $(G.P.)$,where $a, b, c, d$ are in $G.P.$ with common ratio $p$.

If $a, b, c, d$ and $p$ are distinct real numbers such that $(a^2 + b^2 + c^2)p^2 - 2p(ab + bc + cd) + (b^2 + c^2 + d^2) \le 0$,then

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