For  $p,q\in R,$ consider the real valued function  $f\left(x\right)={(x-p)}^2-q,x\in R$  and $q>0.$  Let  $a_1,a_2,a_3$  and $a_4$  be in an arithmetic progression with mean p and positive common difference. If $\left|f\left(a_i\right)\right|=500$  for all  $i=1,2,3,4,$  then the absolute difference between the roots of  $f\left(x\right) = 0$ is _____

$f\left(x\right)=0\Rightarrow {(x-p)}^2-q=0.$ 
Roots are  $p+\sqrt{q},p-\sqrt{q}$  
Absolute difference between roots  $2\sqrt{q}$
Given,  $\left|f\left(a_i\right)\right|=500$
Let  $a_1,a_2,a_3,a_{4}area-3d,a-d,a+d,a+3d$ $\Rightarrow p=a$
$\left|f\left(a_1\right)\right|=500$ $\Rightarrow |9d^2-q|=500$           ……(1)
and  ${\left|f\left(a_1\right)\right|}^2={\left|f\left(a_2\right)\right|}^2$
$\Rightarrow ({(a_1-p)}^2-{(a_2-p)}^2)({(a_1-p)}^2-q+{(a_2-p)}^2-q)=0$
$\Rightarrow \left(10d^2\right)(10d^2-2q)=0\Rightarrow d^2=\frac{q}{5}$
From equation (1) : $\left|\frac{4q}{5}\right|=500$ $\Rightarrow q=625\Rightarrow 2\sqrt{q}=50$