SATHEE: Sequences and Series 3 Question 11

Sequences and Series 3 Question 11

11. If $a, b, c, d$ and $p$ are distinct real numbers such that $(a^{2} + b^{2} + c^{2}) p^{2} - 2 (a b + b c + c d) p$

$+ (b^{2} + c^{2} + d^{2}) \leq 0, then a, b, c, d$

(a) are in $A P$

(b) are in GP

(d) satisfy $a b = c d$

(1987, 2M)

Show Answer

Answer:

Correct Answer: 11. (b)

Solution:

Here, $(a^{2} + b^{2} + c^{2}) p^{2} - 2 (a b + b c + c d) p$

$\begin{matrix} \Rightarrow (a^{2} p^{2} - 2 a b p + b^{2}) + (b^{2} p^{2} - 2 b c p + c^{2}) \\ + (c^{2} p^{2} - 2 c d p + d^{2}) \leq 0 \\ \Rightarrow (a p - b)^{2} + (b p - c)^{2} + (c p - d)^{2} \leq 0 \end{matrix}$

$+ (b^{2} + c^{2} + d^{2}) \leq 0$

[since, sum of squares is never less than zero] Since, each of the squares is zero.

$\begin{array}{ll} ∴ & (a p - b)^{2} = (b p - c)^{2} = (c p - d)^{2} = 0 \\ \Rightarrow & p = \frac{b}{a} = \frac{c}{b} = \frac{d}{c} \end{array}$

$∴ a, b, c, d$ are in GP.

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Sequences and Series 3 Question 11

11. If a,b,c,d and p are distinct real numbers such that (a2+b2+c2)p2−2(ab+bc+cd)p

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11. If $a, b, c, d$ and $p$ are distinct real numbers such that $(a^{2} + b^{2} + c^{2}) p^{2} - 2 (a b + b c + c d) p$