Question

If the difference between the corresponding roots of the equations $x^2+ax+b=0$ and $x^2+bx+a=0$ is same, then

• A. $a+b-4=0$
• B. $a-b+4=0$
• C. $a+b+4=0$
• D. $a-b-4=0$

Answer 1

The correct option is C $a+b+4=0$

If the roots of the equation $a{x}^2+bx+c=0$ are $\alpha$ and $\beta$

Then $\alpha +\beta =\frac{-b}{a}$ ....(1)

and $\alpha \beta =-\frac{c}{a}$ .....(2)

Let ${\alpha }_1,{\beta }_1$ be the roots of $x^2+ax+b=0$ and ${\alpha }_2,{\beta }_2$ be the roots of $x^2+bx+a=0$.

The difference between the corresponding roots is same
$\Rightarrow {\alpha }_1-{\beta }_1={\alpha }_2-{\beta }_2$

The difference between the roots is given by $(\alpha -\beta {)}^2=(\alpha +\beta {)}^2-4\alpha \beta$ ....(3)

Using (1) and (2): ${\alpha }_1+{\beta }_1=-a;{\alpha }_1{\beta }_1=b$

Using 3: $({\alpha }_1-{\beta }_1{)}^2=a^2-4b$ ....(4)

Similarly,

${\alpha }_2+{\beta }_2=-b;{\alpha }_2{\beta }_2=a$

$({\alpha }_2-{\beta }_2{)}^2=b^2-4a$ ....(5)

Now equating (4) and (5)

Therefore, $a^2-4b=b^2-4a$

$\Rightarrow a^2-b^2=4(b-a)$

$\Rightarrow a+b=-4$

$\Rightarrow a+b+4=0$