I can find the fractions:
-0.16666666666666666 → -1/6;  0.8333333333333334 → 5/6;  1.3888888888888888 → 25/18
  [ see here ]
(x + 1/6)² + (y − 5/6)² = 25/18
[or:   3·x² + 3·y² + x − 5·y − 2 = 0]

We can draw the circle with (possiamo tracciare il cerchio con) circle_1.

Let's check also with  (Controlliamo anche con)  R:

source("http://macosa.dima.unige.it/r.R")
PLANE(-2,2,-1,3)
x=c(0,1,-1); y=c(2,1,0)
circle3p(x,y, "red")
POINT(x,y, "blue")
center3p; radius3p
# -0.1666667  0.8333333
#  1.178511
POINT(center3p[1],center3p[2],"blue")

Conic(c(3,0,3,1,-5,-2))
# C1=function(x,y)  3 * x^2 + 0 * x*y + 3 * y^2 + 1 * x + -5 * y + -2 
# circle  xC = -0.1666667  yC = 0.8333333  R = 1.178511 
CUR(C1,"green")
POINT(x,y, "blue")
fraction(c(xC, yC, R^2))
#  -1/6 5/6  25/18