Question

Let $y=a{x}^2+bx+c(a\ne 0)$ and $a,b,c\in R$. If $abc>0$, then which of the following graph(s) satisfy the given condition?

• A.
• B.
• C.
• D.

Answer 1

Answer: (A), (C), (D)

Since, all the given graph are downward, therefore
$a<0$
Let the $x$ coordinate of vertex be $x_1=-\frac{b}{2a}$
$b=-2a{x}_1$
$b$ has same sign as of $x_1$
When $x=0\Rightarrow y=c$
When $abc>0\Rightarrow bc<0$

Case $:1$
$c<0$
$x$ coordinate of vertex $\left(x_1\right)$ is positive.
$b>0$
So,
$bc<0$

Case $:2$
$c<0$
$x$ coordinate of vertex $\left(x_1\right)$ is negative.
$b<0$
So,
$bc>0$

Case $:3$
$c>0$
$x$ coordinate of vertex $\left(x_1\right)$ is negative.
$b<0$
So,
$bc<0$

Case $:4$
$c<0$
$x$ coordinate of vertex $\left(x_1\right)$ is positive.
$b>0$
So,
$bc<0$