Question

The point $P(1,1)$ is translated parallel to $2x=y$ in the first quadrant through a unit distance. Show that the co-ordinates of the new position of $P$ are $(1\pm \frac{1}{\sqrt{5}},1\pm \frac{2}{\sqrt{5}}).$

Answer 1

Any line passing through $P(1,1)$ parallel to $2x=y$ whose

slope$=2=\tan \theta$ is $\frac{x-1}{\cos \theta }=\frac{y-1}{\sin \theta }=r$

or $\frac{x-1}{1/\sqrt{5}}=\frac{y-1}{2/\sqrt{5}}=\pm 1$

$\therefore (x,y)$ is $(1\pm \frac{1}{\sqrt{5}},1\pm \frac{2}{\sqrt{5}}).$