Question

If the roots of the quadratic equation $a{x}^2+bx+c=0$ are opposite in sign and positive root is greater in magnitude, then

• A. $a>0,b>0\text{and}c<0$
• B. $a<0,b<0\text{and}c>0$
• C. $b>0,c>0\text{and}a<0$
• D. $b<0,c<0\text{and}a>0$

Answer 1

Answer: (C), (D)

The correct options are
C $b>0,c>0\text{and}a<0$
D $b<0,c<0\text{and}a>0$

Given equation $a{x}^2+bx+c=0$
Let the roots be $\alpha ,\beta$
Assuming the greater root to be $\beta$
So, $\beta >0\Rightarrow \alpha <0$
Now,
$\alpha +\beta >0\Rightarrow -\frac{b}{a}>0\Rightarrow \frac{b}{a}<0$
So, $a,b$ are of opposite sign.
$\alpha \times \beta <0\Rightarrow \frac{c}{a}<0$
So, $a,c$ are of opposite sign.

Therefore,
$b,c>0\text{and}a<0$
$b,c<0\text{and}a>0$