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

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Question

The given relation can be written as

$\left(a^{2} p^{2}-2 a b p+b^{2}\right)+\left(b^{2} p^{2}+c^{2}-2 b p c\right)+$

$\left(c^{2} p^{2}+d^{2}-2

Vedclass · Accepted Answer

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

Vedclass · Answer

The given relation can be written as

$\left(a^{2} p^{2}-2 a b p+b^{2}\right)+\left(b^{2} p^{2}+c^{2}-2 b p c\right)+$

$\left(c^{2} p^{2}+d^{2}-2 p c d\right) \leq 0$

or $\quad(a p-b)^{2}+(b p-c)^{2}+(c p-d)^{2} \leq 0..........(1)$

since $a, b, c, d$ and $p$ are all real, the inequality $(1)$ is possible only when each of factor is zero

i.e., $a p-b=0, bp-c=0$ and $cp-d=0$

or $\quad p=\frac{b}{a}=\frac{c}{b}=\frac{d}{c}$

or $\quad a, b, c, d$ are in $G.P.$

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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