$The focus of the parabola is (1,1) and the \tan gent at the vertex has the equation x+y=1 which of the following is correct$

$The dis\tan ce from focus to the \tan gent at vertex is a=\sqrt{2}$

$Directrix is a line parallel to the \tan gent at vertex and at a dis\tan ce of 2\sqrt{2} units from focus$

$Directrix is pas\sin g through the image of focus in \tan gent at vertex$

$Hence the equation of the directrix is x+y=0$

$By u\sin g the definition of the parabola, the locus is$

${\left(\frac{x+y}{\sqrt{2}}\right)}^2={\left(x-1\right)}^2+{\left(y-1\right)}^2$

$x^2+y^2+2xy=2x^2+2y^2-4x-4y+4=0$

${\left(x-y\right)}^2=4\left(x+y-1\right)$

$This is the equation of the parabola$

$Vertex is the foot of the perpendicular of focus on the \tan gent at vertex$

$here vertex is \left(\frac{1}{2},\frac{1}{2}\right)$

$a=\frac{1}{\sqrt{2}}, hence the length of the latusrectum is 2\sqrt{2}$