Question

\[a{{x}^{2}}+bx+c=0\]has real and distinct roots \[\alpha \text{ and }\beta \left( \beta >\alpha \right)\]. Further a > 0, b < 0 and c < 0, then
(a) \[0<\beta <\left| \alpha \right|\]
(b) \[0<\left| \alpha \right|<\beta \]
(c) \[\alpha +\beta <0\]
(d) \[\left| \alpha \right|+\left| \beta \right|=\left| \dfrac{b}{a} \right|\]

Answer 1

Hint: In this question, we first need to write the sum of the roots and product of the roots formula. Then by using the given conditions on a, b, c we get the condition on the roots using their sign property. Now, by using the given condition of the roots in the question and their signs we get the result.

Complete step-by-step answer:
As given in the question that \[a{{x}^{2}}+bx+c=0\] has real and distinct roots.

Also given that \[\beta >\alpha \] and \[a>0,b<0,c<0\]

QUADRATIC EQUATION:

If roots of the quadratic equation \[a{{x}^{2}}+bx+c=0\] are \[\alpha \text{ and }\beta \], then

Sum of the roots is given by

\[\alpha +\beta =\dfrac{-b}{a}\]

Product of the roots is given by

\[\alpha \cdot \beta =\dfrac{c}{a}\]

Now, from the given question we get,

\[\Rightarrow \alpha +\beta =\dfrac{-b}{a}\]

As already given in the question that \[a>0,b<0\]

Now, we get that the sum of the roots is greater than 0 as the denominator is greater than 0 and numerator is also greater than 0 by applying the given conditions.

\[\Rightarrow \alpha +\beta >0\]

Let us now consider the product of the roots and apply the given conditions

\[\Rightarrow \alpha \cdot \beta =\dfrac{c}{a}\]

As we already know that \[a>0,c<0\]

Now, by applying these conditions we get that the product of the roots is less than 0 because the numerator is less than 0 and the denominator is greater than 0.

\[\Rightarrow \alpha \cdot \beta =\dfrac{c}{a}<0\]

Now, by considering these two conditions obtained we can say that both the roots are of opposite sign as the product should be less than 0

Also given that \[\beta >\alpha \] so from this we can say that

\[\beta >0,\alpha <0\]

Now, we further get the condition

\[\left| \alpha \right|<\beta \] as \[\alpha +\beta >0\]

Thus, we get the relation \[0<\left| \alpha \right|<\beta \]

Hence, the correct option is (b).


Note:

It is important to note that the relation between the two roots can be obtained by using the sum of the roots condition because the sum should be greater than 0 and both the roots are of opposite sign. Also as given that \[\beta \]is greater than \[\alpha \]we can get the relation.
While getting the signs of the roots we need to use the given conditions and then get the correct relation between the roots because altering the greater than or less than sign changes the result completely.