ISI2014-DCG-22

Question

The conditions on $a$, $b$ and $c$ under which the roots of the quadratic equation $ax^2+bx+c=0 \: ,a \neq 0, \: b \neq 0 $ and $c \neq 0$, are unequal magnitude but of the opposite signs, are the following:

• A. $a$ and $c$ have the same sign while $b$ has the opposite sign.
• B. $b$ and $c$ have the same sign while $a$ has the opposite sign; or $a$ and $b$ have the same sign while $c$ has the opposite sign.
• C. $a$ and $c$ have the same sign.
• D. $a$, $b$ and $c$ have the same sign.

Answer 1

$ax^2+bx+c=0\Rightarrow x^2+\frac{b}{a}x+\frac{c}{a}=0; ~[\because a\ne 0]$

Let $\alpha, \beta$ be the two roots of $x^2+\frac{b}{a}x+\frac{c}{a}=0$.

$$\therefore (x-\alpha)(x-\beta) \equiv x^2+\frac{b}{a}x+\frac{c}{a}=0\\ \Leftrightarrow x^2-(\alpha+\beta)x+\alpha\beta \equiv x^2+\frac{b}{a}x+\frac{c}{a}\\ \Leftrightarrow \frac{b}{a}=-(\alpha+\beta),~ \frac{c}{a}=\alpha\beta$$

Since the roots are unequal and of opposite sign, their product must be negative.

So $$\alpha\beta<0\\ \Rightarrow \frac{c}{a}<0\\ \Rightarrow c\cdot \frac{1}{a}<0$$

It means $c$ and $a$ must be of opposite sign. Only the option B satisfies this statement.

So the correct answer is B.

Answer 2

Let us take a example.let two roots be 2, -3.

sum of roots=-1,product=$-6$

Therefore, equation is $x^{2}+x-6$.

This equation eliminates options A,C,D.

Here a,b have same sign while c have different sign.

If roots are -2,3.

sum of roots=1,product of roots=$-6$.

Therefore, equation is $x^{2}-x-6$.

Here b,c have same sign while a has opposite sign.

Therefore option B is right answer.

OR

for the equation$ax^{2}+bx+c$$=0$,

Sum of roots =$-b/a$ and product of roots=$c/a$

Roots are of opposite sign$\Rightarrow$ product is negative and sum of roots is either negative or positive(see above example).

CASE-1:Sum of roots is -ve and product of roots is -ve

$\Rightarrow$  -b/a<0,c/a<0

$\Rightarrow$ b/a>0,c/a<0

if a is +ve, then b is +ve,c is -ve.

if a is -ve, then b is -ve,c is +ve.

Therefore, a,b has same sign while c has opposite sign.

CASE-2: Sum of roots is +ve and product of roots is -ve.

$\Rightarrow$  -b/a>0,c/a<0

$\Rightarrow$ b/a<0,c/a<0

if a is +ve, then b is -ve,c is -ve.

if a is -ve, then b is +ve,c is +ve.

Therefore, b,c has same sign while a has opposite sign.

option B is the answer.