Question

The graph of a quadratic polynomial $y=a x^2+b x+c(a, b, c \in R)$ with vertex on y-axis is as shown in the figure. Then which one of the following statement is CORRECT?

• A. Product of the roots of the corresponding quadratic equation is positive.
• B. Discriminant of the quadratic equation is negative.
• C. Nothing definite can be said about the sum of the roots, whether positive, negative or zero.
• D. Both roots of the quadratic equation are purely imaginary.

Answer 1

Answer: (A), (B), (D)

Roots are purely imaginary
i.e. $ i \beta$ and $-i \beta$
$\therefore $ sum of roots $=0$
incorrect (C)
product of roots $=-i^2 \beta^2=\beta^2 $
$\Rightarrow $ product $>0 ; \frac{ c }{ a }>0$
$ \Rightarrow c =+ ve$ note that $-\frac{b}{2 a}=0 \Rightarrow b=0$
hence $y=a x^2+c$ when $x=0, y=c>0 $
$ \Rightarrow y=a x^2+c$ where $c>0$