Question

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

• A. A.P.
• B. H.P.
• C. G.P.
• D. $ab=bc$

Answer 1

The correct option is C G.P.

Given, $(a^2+b^2+c^2)x^2-2(ab+bc+cd)x+(b^2+c^2+d^2)\le 0$

$(a^2{x}^2-2abx+b^2)+(b^2{x}^2-2bcx+c^2)+(c^2{x}^2-2dcx+d^2)\le 0$ $\Rightarrow (ax-b{)}^2+(bx-c{)}^2+(cx-d{)}^2\le 0$

$\Rightarrow (ax-b{)}^2=0,(bx-c{)}^2=0$, and $(cx-d{)}^2=0$

$\Rightarrow x=\frac{b}{a}=\frac{c}{b}=\frac{d}{c}$

$\Rightarrow a,b,c,d$ are in G.P.

Also, $\frac{b}{a}=\frac{d}{c}$

$\Rightarrow ad=bc$