LineCircle.ex

defmodule LineCircle do
  @type point() :: {number, number}
  @spec intersection([point()], point(), number()) :: [point()] | nil
  def intersection([{x1, y1}, {x2, y2}], {cx, cy}, r) when is_number(r) do
    dx = x2 - x1
    dy = y2 - y1
    dr = :math.sqrt(dx * dx + dy * dy)
    matrix = x1 * y2 - x2 * y1

    discriminant = square(r) * square(dr) - square(matrix)

    if discriminant < 0 do
      nil
    else
      x_1 = (matrix * dy + sign(dy) * dx * :math.sqrt(discriminant)) / (dr * dr)
      x_2 = (matrix * dy - sign(dy) * dx * :math.sqrt(discriminant)) / (dr * dr)
      y_1 = (-matrix * dx + abs(dy) * :math.sqrt(discriminant)) / (dr * dr)
      y_2 = (-matrix * dx - abs(dy) * :math.sqrt(discriminant)) / (dr * dr)
      [{x_1 + cx, y_1 + cy}, {x_2 + cx, y_2 + cy}]
    end
  end

  defp square(x), do: x * x

  defp sign(x) do
    if x < 0 do
      -1
    else
      1
    end
  end
end

LineCircleTest.exs

   test "sholder" do
      a = {3575.2, 2526.5}
      b = {2567.2, 2526.5}
      center = {2600.24, 2601.4}
      radius = 52 * 72
      assert [p1, p2] = LineCircle.intersection([a, b], center, radius)|> IO.inspect()
      assert y(p1) == 2526.5
      # Y == 2127.9 here
    end