Equation of the directrix of the parabola whose focus is ( 0 , 0 ) and the tangent at the vertex is x - y +…

We have, $x-y+1=0 ...\left(\mathrm{i}\right)$

Focus is $(0,0)$.

Perpendicular distance from $(0,0)$ to $\left(\mathrm{i}\right)$ is given by

$d=\left|\frac{0-0+1}{\sqrt{1^2+1^2}}\right|=\frac{1}{\sqrt{2}}$

Now, equation of directrix is given by

$x-y+c=0$  [ $\because$ directrix is parallel to tangent at vertex ]

Perpendicular distance from $(0,0)$ to directrix is

$\left|\frac{0-0+c}{\sqrt{1^2+1^2}}\right|=2\left(\frac{1}{\sqrt{2}}\right)\Rightarrow \frac{c}{\sqrt{2}}=\frac{2}{\sqrt{2}}\Rightarrow c=2$

Hence, equation of directrix is $x-y+2=0$.