If a, b, c, d and p are distinct real numbers such that (a2+b2+c2) p2+2(a b+b c+c d) p+ (b2+c2+d2) ≤ 0, then a, b, c, d

Question

If a, b, c, d and p are distinct real numbers such that (a2+b2+c2) p2+2(a b+b c+c d) p+ (b2+c2+d2) ≤ 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

Tardigrade Admin · Accepted Answer

Write the expression as (a p+b)2+(b p+c)2+(c p+d)2 ≤ 0 . ⇒ a p+b=0, b p+c=0, c p+d=0 ⇒ (b/a)=(c/b)=(d/c)=-p .

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

are in A.P.

are in G.P.

are in H.P.

none of these

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

Q. If a,b,c,d and p are distinct real numbers such that (a2+b2+c2)p2+2(ab+bc+cd)p+(b2+c2+d2)≤0, then a,b,c,d