Question

If $a,b,c>0,a^2=bc$ and a + b + c = abc, then least value of $a^4+a^2+7$ is $‘{\alpha }^{\prime }$, the value of $\alpha -16$ is___

• A. 3

Answer 1

The correct option is A

3

$bc=a^2$
b + c = abc - a
$=a^3-a$
$\Rightarrow$ ‘b’ and ‘c’ are roots of equation
$x^2-(b+c)x+bc=0$
i.e. $x^2-(a^3-a)x+a^2=0$
$\because$ ‘b’ and ‘c’ are real
$D\ge 0(a^3-a{)}^2-4a^2\ge 0\Rightarrow a^2\ge 3\Rightarrow a^4+a^2+7\ge 9+3+7a^4+a^2+7\ge 19\Rightarrow \alpha =19\alpha -16=3$