Question

If $a,b,c,d,x$ are distinct 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. $A.G.P$
• C. $H.P$
• D. $G.P$

Answer 1

The correct option is D $G.P$

The inequality $(a^2+b^2+c^2)x^2-2(ab+bc+ca)x+(b^2+c^2+d^2)\le 0$ can be written as ${(ax-b)}^2+{(bx-c)}^2+{(cx-d)}^2\le 0$
Which is possible only if $ax-b=0i.e.x=\frac{b}{a}$
& ${(bx-c)}^2=0i.ex=\frac{c}{b}$
& ${(cx-d)}^2=0i.ex=\frac{d}{c}$
$\therefore \frac{b}{a}=\frac{c}{b}=\frac{d}{c}=x$
$\Rightarrow a,b,c,d$ are in $G.P$
Hence (d) is correct answer.