Explanation

(c)
Given equations are comparing with $ax^{2} + bx + c = 0$
And $a^{'}x^{2} + b^{'}x + c^{'} = 0$ respectively, we get
$a = 1,\ \ \ \ b = 2a,\ \ \ c = a^{2} - 1$
And $a^{'} = 1,\ \ \ \ \ b^{'} = 2b,\ \ \ \ c^{'} = b^{2} - 1$
Condition for common roots is
$\left( ac^{'} - a^{'}c \right)^{2} = \left( bc^{'} - b^{'}c \right)\left( ab^{'} - a^{'}b \right)$
$\Rightarrow \ \ \left\lbrack 1\left( b^{2} - 1 \right) - 1\left( a^{2} - 1 \right) \right\rbrack^{2} = \left\lbrack 2a\left( b^{2} - 1 \right) - 2b\left( a^{2} - 1 \right) \right\rbrack\left\lbrack 1(2b) - 1(2a) \right\rbrack$
$\Rightarrow \ \ \ \left( b^{2} - a^{2} \right)^{2} = 4(b - a)(b - a)(ab + 1)$
$\Rightarrow (b + a)^{2} = 4(ab) + 4$
$\Rightarrow \ \ (b - a)^{2} = 4$
$\Rightarrow \ \ a - b = 2$