Given,

A plane $P$ has the points $\hat{\mathrm{i}}, \hat{\mathrm{j}}$ and $\hat{\mathrm{i}}+ \hat{\mathrm{j}}+\hat{\mathrm{k}}$

Now let equation of plane be $ax+by+cx+d=0$,

Now satisfying the given point one by one we get,

$a+d=0, b+d \& a+b+c+d=0$

$\Rightarrow a=-d, b=-d \& c=d$

Now putting the value in equation we get equation of plane as $x+y-z-1=0$

Now direction ratio of normal to plane is $\hat{\mathrm{i}}+ \hat{\mathrm{j}}-\hat{\mathrm{k}}$

So equation of line passing through $\left(3,0,5\right)$ with direction ratio $\left(1,1,-1\right)$ will be $\frac{x-3}{1}=\frac{y}{1}=\frac{z-5}{-1}$