For three consecutive odd integers a ,   b and c , if the variable line a x + b y + c = 0 always passes…

The given line is $ax+by+c=0$. 

Since $a, b$ and $c$ are three consecutive odd integers, so they are in arithmetic progression,

$\Rightarrow b-a=c-b$

$\Rightarrow c=2b-a$

Hence, the equation of line is $ax+by+2b-a=0$

$\Rightarrow a\left(x-1\right)+b\left(y+2\right)=0$.

Now, the is given that the line always pass through point $\left(\alpha , \beta \right)$

$\Rightarrow a\left(\alpha -1\right)+b\left(\beta +2\right)=0$

This is only possible when, $\left(\alpha -1\right)=0$ and $\left(\beta +2\right)=0$

$\Rightarrow \alpha =1$ and $\beta =-2$

$\Rightarrow {\alpha }^2+{\beta }^2=1^2+{\left(-2\right)}^2=5$