the nearest point on the line 2x-y+5=0 from the origin is
 

Hello Piyush, the nearest point on line 2x - y + 5 = 0 from origin (0,0) is the nothing but the foot of perpendicular drawn from origin to that line
Slope of the given line is -2/-1 = 2 [ simply follow the technique slope = - coeff of x / coeff of y]
So slope of any line perpendicular to the given line has to be -1/2 (since m₁* m₂ = - 1)
Now equation of perpendicular line passing through origin (0,0)
y - 0 = -1/2 (x - 0)
===> x + 2 y = 0
Point of intersection of this line and given line is the required nearest point
Solving we get (-2,1)
(-2,1) is the nearest point on the line