A movable parabola touches the x-axis and the y-axis at (1,0 ) and (0,1). Then the locus of the focus of the parabola is 

Let S= $(\alpha ,\beta )$
EQ is Directrix is $y=\frac{-\beta }{\alpha }x\Rightarrow \alpha y+\beta y=0$ 
Let (x,y) be a pt on parabola  $s{p}^2=p{M}^2$
$\Rightarrow {(x-\alpha )}^2+{(y-\beta )}^2={\left|\frac{\alpha y+\beta x}{\sqrt{{\alpha }^2+{\beta }^2}}\right|}^2,$

Its passing through (1,0) 
 $\Rightarrow (1-{\alpha }^2)+{\beta }^2=\frac{{\alpha }^2}{{\alpha }^2+{\beta }^2}---\left(1\right)$
Passing through (0,1)
$\Rightarrow {\alpha }^2+{(\beta -1)}^2=\frac{{\beta }^2}{{\alpha }^2+{\beta }^2}---\left(2\right)$