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Q)

$a,b\in R$,roots of $x^2+ax+b=0$ are $\alpha,\beta$ and roots of $x^2+bx+a=0$ are $\alpha+k,\beta+k$ where $k\neq 0$.Which of the following is true?

$\begin{array}{1 1}(A)\;a+b+1=0&(B)\;a=b+1\\(C)\;a+b+4=0&(D)\;\text{None}\end{array} $

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A)
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$\alpha+\beta=-a$
$\alpha\beta=b$
$\alpha+\beta+2k=-b\Rightarrow k=\large\frac{a-b}{2}$
$(\alpha+k)(\beta+k)=a$
$\Rightarrow \alpha\beta+k(\alpha+\beta)+k^2=a$
$\Rightarrow b+k(-a)+k^2=a$
$\Rightarrow b-a(\large\frac{a-b}{2}+(\large\frac{a-b}{2})^2$$=a$
$\Rightarrow a+b+4=0$
Hence (C) is the correct answer.
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