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  $\mathrm{S}=(\mathrm{\alpha },\mathrm{\beta })$ be locus of focus
Equation of idirectrix is $\mathrm{y}=\frac{-\mathrm{\beta }}{\mathrm{\alpha }}\mathrm{x}\Rightarrow \mathrm{\alpha y}+\mathrm{\beta x}=0,$
Let (x, y) be a point on parabola  then $S{P}^2=P{M}^2$
$\Rightarrow (\mathrm{x}-\mathrm{\alpha }{)}^2+(\mathrm{y}-\mathrm{\beta }{)}^2={\left|\frac{\mathrm{\alpha y}+\mathrm{\beta x}}{\sqrt{{\mathrm{\alpha }}^2+{\mathrm{\beta }}^2}}\right|}^2$
 

Its passing through (1,0)
$\Rightarrow {\left(1-{\mathrm{\alpha }}^{ }\right)}^2+{\mathrm{\beta }}^2=\frac{{\mathrm{\alpha }}^2}{{\mathrm{\alpha }}^2+{\mathrm{\beta }}^2}---\left(1\right)$
Passing through (0,1),
$$\begin{gathered} \Rightarrow {\mathrm{\alpha }}^2+(\mathrm{\beta }-1{)}^2=\frac{{\mathrm{\beta }}^2}{{\mathrm{\alpha }}^2+{\mathrm{\beta }}^2}---(2) \\ By adding (1) and (2) \\ {(1-\alpha )}^2+{\beta }^2+{\mathrm{\alpha }}^2+(\mathrm{\beta }-1{)}^2=1 \end{gathered}$$

$locus is 2x^2+2y^2-2x-2y+1=0$