Question

If the roots of $a{x}^2+bx+c=0$ are equal in magnitude but opposite in sign, then a possible condition is

• A. $a<0,c<0$
• B. $a<0,c>0,b>0$
• C. $a>0,b=0,c<0$
• D. $a>0,b=0,\underset{–}{c}>0$

Answer 1

The correct option is C $a>0,b=0,c<0$

$\Rightarrow$ The given quadratic equation is $a{x}^2+bx+c=0$

$\Rightarrow$ Let two roots be $\alpha$ and $\beta$.

$\Rightarrow$ $\alpha =-\beta$ [ Given ]

$\Rightarrow$ $\alpha +\beta -\frac{-b}{a}$

$\Rightarrow$ $-\beta +\beta =\frac{--b}{a}$

$\Rightarrow$ $0=\frac{-b}{a}$

$\therefore$ $b=0$ ------- ( 1 )

We have one root negative so,

$\therefore$ $\alpha \beta <0$

$\therefore$ $\frac{c}{a}<0$

So, here either $c$ or $a$ will be negative.

Means, $a$ and $c$ will be having opposite sign.

$a>0,c<0$ or $c>0,a<0$ and $b=0$.

In options we can see option $C$ satisfy all the conditions.

$\therefore$ Posibble conditions are $a>0,b=0,c<0$