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Q.
The difference between the corresponding roots of $x^{2} + ax + b = 0$ and $x^{2} + bx + a = 0$ is same and $a\ne b,$ then
Complex Numbers and Quadratic Equations
Solution:
Let $\alpha,\beta$ and $\gamma,\delta$ be the roots of the equations
$x^{2}+ ax + b = 0$ and $x^{2}+ bx +a =0,$ respectively
Therefore,
$\alpha+\beta=-a, \alpha\beta=b$
and $\delta+\gamma=-b, \gamma\delta=a.$
Given $\left|\alpha-\beta\right|=\left|\gamma-\delta\right| $
$\Rightarrow \left(\alpha+\beta\right)^{2} -4\alpha\beta=\left(\gamma+\delta\right)^{2}-4\gamma\delta$
$\Rightarrow a^{2} -4b =b^{2} =4a$
$\Rightarrow \left(a^{2}=b^{2}\right)+4 \left(a-b\right)=0$
$\Rightarrow a+b+4=0 \,\,\,\left(\because a\ne b\right)$