Question

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

• A. are in A.P.
• B. are in G.P.
• C. are in H.P.
• D. none of these

Answer 1

Answer: (B)

Write the expression as
$(a p+b)^2+(b p+c)^2+(c p+d)^2 \leq 0 . $
$\Rightarrow a p+b=0, b p+c=0, c p+d=0 $
$\Rightarrow \frac{b}{a}=\frac{c}{b}=\frac{d}{c}=-p$ .