Question

The graph of a quadratic polynomial $y=a{x}^2+bx+c(a,b,c\in R)$ 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 $-\frac{b}{2a}=0\Rightarrow b=0$
hence $y=a{x}^2+c$ when $x=0,y=c>0$
$\Rightarrow y=a{x}^2+c$ where $c>0$