Question

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

• A. are in $A.P.$
• B. are in $G.P.$
• C. are in $H.P.$
• D. satisfy $ab=cd$

Answer 1

Answer: (B)

are in $G.P.$

We have, $(a^2+b^2+c^2)p^2-2(ab+bc+cd)p+(b^2+c^2+d^2)\le 0$

$\Rightarrow {(ap-b)}^2+{(bp-c)}^2+{(cp-d)}^2\le 0\Rightarrow {(ap-b)}^2+{(bp-c)}^2+{(cp-d)}^2=0(\because a,b,c,d,p\in R)\Rightarrow ap-b=0,bp-c=0,cp-d=0$

$\Rightarrow \frac{b}{a}=\frac{c}{b}=\frac{d}{c}=p\Rightarrow a,b,c,d$ are in $G.P.$