Question

The point $(1,1)$ is translated parallel to $y=2x$ in the first quadrant through unit distance. What will be its co-ordinates in the new position ?

Answer 1

Any line through $(1,1)$ parallel to $y=2x$ is
$y-1=2(x-1)$
$\tan \theta =2\therefore \cos \theta =\frac{1}{\sqrt{5}}$, $\sin \theta =\frac{2}{\sqrt{5}}$.
Now we have to find a point on this line which is at a unit distance from $(1,1)$.
$\therefore \frac{x-1}{\cos \theta }=\frac{y-1}{\cos \theta }=r=\pm 1$
$\therefore x=\cos \theta +1,y=\sin \theta +1$
or $(x=-\cos \theta +1,y=-\sin \theta +1)$
or $(1+\frac{1}{\sqrt{5}},1+\frac{2}{\sqrt{5}});(1-\frac{1}{\sqrt{5}},1-\frac{2}{\sqrt{5}})$
Both lie in $Ist$ quadrant.