Step 1
The coordinates of the arbitrary point on $x+y=4$ can be obtained by putting $x=t$ (or $y=t)$ and then obtaining $y$(or $ x)$ from the equation of the line, where $t$ is a parameter.
Substituting $x=t$ in $x+y=4$ we get
$y=4-t$
$ \therefore $ The coordinates of an arbitrary point on the given line are $P(t, 4-t).$
Let $P(t, 4-t)$ be the required point.
Step 2
It is given that the distance of $P$ from the line $ 4x+3y-10=0$ is unity.
$ \therefore \bigg| \large\frac{4t+3(4-t)-10}{\sqrt{4^2+3^2}} \bigg|$$=1$
(i.e) $|t+2|=5$
(i.e) $t+2= \pm 5$
$ \therefore t = -7\: or \: t = 3$
Hence the required points are (-7,11) or ( 3,1)