If α ,   β are natural numbers such that 100 α - 199 β = ( 100 ) ( 100 ) + ( 99 ) (…

Let $S=\left(100\right)\left(100\right)+\left(99\right)\left(101\right)+\left(98\right)\left(102\right)+...+\left(2\right)\left(198\right)+\left(1\right)\left(199\right)$

$S=\sum _{x=0}^{99}\left(100-x\right)\left(100+x\right)=\sum _{x=0}^{99}{100}^2-\sum _{x=0}^{99}{x}^2$

$S={100}^3-\frac{99\times 100\times 199}{6}={100}^3-199\times 1650$

Given that, $S={100}^{\alpha }-199\beta$

$\therefore \alpha =3$, $\beta =1650$

The slope of line passing through $\left(\alpha ,\beta \right) \& \left(0,0\right)$$=\frac{1650-0}{3-0}=550$