issacnewton said:
fresh_42, is ##d## the perpendicular distance from the line ##y = (2/5)x + c## and the center of the circle ? I don't think I understand what you are saying. My last post was reply to pasmith's post. sorry for misunderstanding.
Yes. The perpendicular is the shortest distance between the line and the circle. The problem asks for a specific radius ##r_0## such that ...
issacnewton said:
... any line with slope ##(2/5)## intersects some of these circles.
That means, regardless of the choice of ##c## (=any line with slope ##0.4##), i.e. for all values ##c,## the line has to intersect at least one circle (=some of these circles), i.e. there exists at least one pair ##(n,m)## such that the (red) distance ##d=d(c,n,m,r)## between the line and the circle is less than the radius ##r## in order to intersect it.
This means in a formula: Find
$$
r_0=\min\{r\geq 0\,|\,\forall\,c\in \mathbb{R}\,\exists\, (n,m)\in \mathbb{Z}^2\, : \,d(c,n,m,r)\leq r\}
$$
The first step is to determine this distance ##d.##
The second step is to examine under which conditions ##d(c,n,m,r)\leq r## has always a solution, no matter what the parameter ##c## is. This means in particular that we cannot have roots of negative numbers.
The last step is to determine the minimal value ##r## such that this condition always holds.
So whatever you do, we need to know ##d=d(c,n,m,r),## the length of the red line, the length between the point ##C=(n,m)## and the point ##P## on the straight ##L\, : \,y=0.4\,x+c## that is perpendicular to the circle. I would start by finding the equation that describes the red line, then calculate the point ##P## on ##L## that is perpendicular to the circle with center at ##C=(n,m)## and radius ##r,## then determine ##d=d(c,n,m,r)= |\overline{PC}|## and analyze the condition ##d\leq r.##