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Algebra of Derivatives

If a,b,c,d ...

Question

If $a, b, c, d$ and $p$ are distinct real numbers such that $(a^{2} + b^{2} + c^{2}) p^{2} - 2 (a b + b c + c d) p + (b^{2} + c^{2} + d^{2}) \leq 0$ then $a, b, c$ and $d$

are in

A . P .

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are in

G . P .

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are in

H . P .

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satisfy

a b = c d

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Solution

The correct option is A are in $G . P .$
We have, $(a^{2} + b^{2} + c^{2}) p^{2} - 2 (a b + b c + c d) p + (b^{2} + c^{2} + d^{2}) \leq 0$
$\Rightarrow {(a p - b)}^{2} + {(b p - c)}^{2} + {(c p - d)}^{2} \leq 0 \Rightarrow {(a p - b)}^{2} + {(b p - c)}^{2} + {(c p - d)}^{2} = 0 (∵ a, b, c, d, p \in R) \Rightarrow a p - b = 0, b p - c = 0, c p - d = 0$
$\Rightarrow \frac{b}{a} = \frac{c}{b} = \frac{d}{c} = p \Rightarrow a, b, c, d$ are in $G . P .$

Suggest Corrections

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