Question

For any three rational numbers $a,b,c$ and $a+b+c=0;a,c>0.$ Now, consider the parabola represented by the quadratic expression $y=a{x}^2+bx+c.$ Then select the correct options.

• A. Vertex lies in $I^{st}$ quadrant
• B. Vertex lies in $I{V}^{th}$ quadrant
• C. If $a=c\Rightarrow$ vertex lies on $x$-axis
• D. $x=1$ is one of the roots of the equation

Answer 1

Answer: (B), (C), (D)

$x=1$ is one of the roots of the equation

Given : $y=a{x}^2+bx+c;a+b+c=0;$
$a>0,c>0;a,b,c\in R$

$\because a+b+c=1\Rightarrow x=1$ is one of the roots of the equation.

Product of roots $=\frac{c}{a}>0$
$\therefore$ 2nd root of the equation will also be positive.
Thus the graph will look like this:


From the graph we can see that the vertex of the parabola lies on the $I{V}^{th}$ quadrant

Now, if the vertex lie on the $x-$ axis then the $y-$ coordinate of the vertex is equal to $0.$

We have $y$-coordinate is $0$

$\Rightarrow -\frac{D}{4a}=0$

$\Rightarrow -\frac{b^2-4ac}{4a}=0$

$\Rightarrow -\frac{(-(a+c){)}^2-4ac}{4a}=0$

$\Rightarrow -\frac{(a-c{)}^2}{4a}=0\Rightarrow (a-c{)}^2=0\Rightarrow a=c$