Recent content by issacnewton

I have found solution on some site which I will explain here. As I have already shown that if the line ##y = (2/5)x + c## intersects the circle ##(x-n)^2 + (y-m)^2 = r^2##, then it will also intersect the circle ##(x-n-5k)^2 + (y -m-2k)^2 = r^2##, where ##k \in \mathbb{Z}##. Now, consider the...

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.

Ok, maybe I made a mistake. After plugging ##y = (2/5)x + c## into ##(x-n)^2 + (y - m)^2 = r^2##, I get the following quadratic equation in ##x## $$ \frac{29}{25}x^2 + \left[ -2n + \frac{4}{5}(c-m) \right] x + n^2 + (c-m)^2 - r^2 = 0 $$ As suggested by pasmith, tangent occurs when there is...

If ##y = (2/5)x + c## is tangent to the circle, then the distance from the center ##(n,m)## to the tangent is given by the following formula from geometry $$ d = \frac{|2n - 5m + 5c|}{\sqrt{2^2 + (-5)^2}} = \frac{|2n - 5m + 5c|}{\sqrt{29}} $$

can anybody guide me further in this problem ?

Ok, let ##(x_1, y_1)## be the point of intersection of the line ##y = (2/5)x + c## and the circle of radius ##r## with center at ##(n, m)##. The equation of such a circle is $$ (x - n)^2 + (y - m)^2 = r^2 $$ Since ##(x_1, y_1)## also lies on this circle, we have $$ (x_1 - n)^2 + (y_1 - m)^2 =...

should all circles have the same radius ?

fresh_42, I can't understand how pasmith came about point ##(n+5k,m+2k)##. So, could not follow the hint

I am saying that if the line is ##y = 1/2##, then the minimum radius of the circles should be ##r = 1/2## so that this line touches infinitely many circles at tangent point. I have no idea how to go about a general line like ##y = mx + c##.

for horizontal lines (say y = 1/2), I can see that minimal radius is ##1/2##, so that lines only are tangent to the circles with centers at the lattice points. But how do we generalize this to the lines of the form ##y = mx + c## ?

How did you come about the point ##(n + 5k, m + 2k)## ?

So, the way I understand this problem, I think the line ##y = (2/5)x + c## should only intersect some of the circles drawn around the lattice points. But, I am not sure I even understand the problem statement. Can the line pass through the lattice points ? My first goal is to understand the...

This makes sense.

Ok, so any real number can be represented as a cut (say ##\alpha##), which is a subset of ##\mathbb{Q}## with the following three properties (I) ##\alpha \ne \varnothing##, and ##\alpha \ne \mathbb{Q}## (II) If ##p \in \alpha##, ##q \in \mathbb{Q}##, and ##q < p##, then ##q \in \alpha## (III)...

I am learning analysis from Rudin's famous book (baby rudin). I am confused about how ##\mathbb{R}## is defined in this book. In the appendix of chapter 1, he says that members of ##\mathbb{R}## will be certain subsets of ##\mathbb{Q}##, called cuts. Is this definition different from the way we...