In the xy- coordinate plane, how many points are a distance of 4 units from the origin?

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In the xy- coordinate plane, how many points are a distance of 4 units from the origin?

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Answer 1 · 2015-05-30T21:45:02

The question boils down to:
How many solutions are there for $x^2+y^2 = 4^2$ ?

There are $4$ solutions with $x, y in ZZ$ namely $(0, 4)$ , $(0, -4)$ , $(4, 0)$ , $(-4, 0)$

There are an infinite number of solutions with $x, y in RR$ , namely
$(4cos theta, 4sin theta)$ for $0 <= theta < 2pi$

How about solutions with $x, y in QQ$ ?

For any non-negative integer $k$ , define

$a_k = 2k+3$

$b_k = (a_k^2-1)/2 = 2k^2+6k+4$

$c_k = (a_k^2+1)/2 = 2k^2+6k+5$

Then $a_k$ , $b_k$ and $c_k$ are the lengths of the sides of a right angled triangle and they are integers, e.g. $(3,4,5)$ , $(5,12,13)$ , $(7,24,25)$ .

Then the point $((4a_k)/c_k, (4b_k)/c_k)$ satisfies $x^2+y^2=4$ and $x, y in QQ$ .

Since there are a countably infinite number of choices for $k$ , there are an infinite number of points $(x, y)$ with rational coordinates at a distance of $4$ from the origin.

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In the xy- coordinate plane, how many points are a distance of 4 units from the origin?

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