Question

A line is drawn through $P(3,4)$ inclined to $x-$axis at an angle $3\pi /4$. Find its equation and also the co-ordinates of two points on it on opposite sides of $P$ at a distance $\sqrt{2}$ from it.

Answer 1

Line is $\frac{x-3}{\cos \left(3\pi /4\right)}=\frac{y-4}{\sin \left(3\pi /4\right)}=r$, say
or $\frac{x-3}{-1/\sqrt{\left(2\right)}}=\frac{y-4}{1/\sqrt{\left(2\right)}}=r$
Its equation is $x-3=-(y-4)$
or $x+y-7=0$.
Any point on it is $(-r/\sqrt{\left(2\right)}+3r/\sqrt{\left(2\right)}+4)$
Since the two points are at a distance $\sqrt{2}$ from $P$ on opposite sides,
hence choosing $r=\pm \sqrt{2}$, we et the points as
$(-\sqrt{\left(2\right)}.1/\sqrt{\left(2\right)}+3,\sqrt{\left(2\right)}.1/\sqrt{\left(2\right)}+4)$
and $(\sqrt{\left(2\right)}.1/\sqrt{\left(2\right)}+3,-\sqrt{\left(2\right)}.1/\sqrt{\left(2\right)}+4)$
or $(2,5)$ and $(4,3)$