Question

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

• A. are in A.P.
• B. are in G.P.
• C. are in H.P.
• D. satisfy a b=c d

Answer 1

Answer: (B)

We have, $\left(a^2+b^2+c^2\right)p^2-2(ab+bc+cd)p+\left(b^2+c^2+d^2\right)\le 0$
$\Rightarrow (ap-b{)}^2+(bp-c{)}^2+(cp-d{)}^2\le 0$
$\Rightarrow (ap-b{)}^2+(bp-c{)}^2+(cp-d{)}^2=0$
$(a,b,c,d,p\in R)$
$\Rightarrow ap-b=0,$
$bp-c=0,cp-d=0$
$\Rightarrow \frac{b}{a}=\frac{c}{b}=\frac{d}{c}=p$
$\Rightarrow a,b,c,d$ are in G.P.