If the length of the latus rectum of a parabola, whose focus is a , a and the tangent at its vertex is x + y…

Given the equation of tangent at vertex is $x+y=a$

and the focus is $\left(a,a\right)$

Now the perpendicular distance from focus to tangent is 

$\frac{\left|a+a-a\right|}{\sqrt{2}}=\frac{\left|a\right|}{\sqrt{2}}$

Also given length of latus rectum$=16$

i.e. $\left|\frac{a}{\sqrt{2}}\right|=\frac{16}{4}=4$

$\Rightarrow \left|a\right|=4\sqrt{2}$