Question

Graph of $f ( x )= ax ^2+ bx + c , a , b , c \in R , a \neq 0$ is shown in the adjacent figure. If area of rectangle $O A B C=18$ sq. units ( $B$ is the vertex of the parabola) and sum of the roots of the equation $f(x)$ $=0$ is 12 , then

• A. least value of ac is $-\frac{1}{16}$
• B. least value of ac is $\frac{1}{2}$
• C. if $a c=\frac{1}{2}$ then value of $b$ is -1
• D. if $ac =\frac{1}{2}$ then value of $b$ is 2 .

Answer 1

Answer: (C)

$\frac{-b}{a}=12 $
$\Rightarrow \left(\frac{-D}{4 a}\right) \cdot\left(\frac{-b}{2 a}\right)=18$
$\Rightarrow \left(\frac{-D}{4 a}\right) \cdot\left(\frac{12}{2}\right)=18$
$\Rightarrow \frac{-\left(b^2-4 a c\right)}{a}=12 $
$\Rightarrow -b^2+4 a c=-b$
$\Rightarrow a c=\frac{b^2-b}{4}$
$\Rightarrow ac _{\min }=\frac{\left( b -\frac{1}{2}\right)^2-\frac{1}{4}}{4}=\frac{-1}{16}$
which is not possible as a and $c$ both are positive.
$\text { If } a c=\frac{1}{2} $
$ -b^2+2=-b$
$\Rightarrow b^2-b-2=0$
$ b=-1,2$
$b \neq 2 \Rightarrow a=\frac{-1}{6}, a>0$