Question

lf the difference of the roots of the equation $x^2-bx+c=0$ is equal to the differecne of the roots of the equation $x^2-cx+b=0$ and $b\ne c$, then $b+c=$

• A. $0$
• B. $2$
• C. $4$
• D. $-4$

Answer 1

The correct option is D $-4$

Let $(\alpha ,\beta )$ and $(\gamma ,\delta )$ be the roots of the first equation and second equation respectively.

Then, for the first equation

$\alpha +\beta =b$ and $\alpha \beta =c$

Now $(\alpha +\beta {)}^2=b^2$

$\Rightarrow (\alpha -\beta {)}^2+4\alpha \beta =b^2$

$\Rightarrow (\alpha -\beta {)}^2=b^2-4\alpha \beta$

$\Rightarrow |\alpha -\beta |=\sqrt{b^2-4c}$

Similarly for the second equation

$|\gamma -\delta |=\sqrt{c^2-4b}$

As per the given condition,

$\sqrt{b^2-4c}=\sqrt{c^2-4b}$

$\Rightarrow b^2-4c=c^2-4b$

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

$\Rightarrow (b+c)(b-c)=-4(b-c)$

Therefore, option D is correct.