Question 1132621
The intercepts with the axes are easy to find.
 
The x-axis is the line {{{y=0}}} , so we find the intercepts from
{{{system(y=0,3x^2+3y^2-5x+3y=0)}}} --> {{{system(y=0,3x^2-5x=0)}}} --> {{{system(y=0,x(3x-5)=0)}}} --> {{{system(y=0,"and",x=0,"or",x=5/3)}}} .
Those intercepts are the points {{{O(0,0)}}} and {{{A(5/3,0)}}} .
The distance between the two intercepts of the circle with the x-axis is
{{{5/3-0=highlight(5/3)}}} .
 
The y-axis is the line {{{x=0}}} , so we find the intercepts from
{{{system(x=0,3x^2+3y^2-5x+3y=0)}}} --> {{{system(x=0,3y^2+3y=0)}}} --> {{{system(x=0,3y(y+1)=0)}}} --> {{{system(x=0,"and",y=0,"or",y=-1)}}} .
Those intercepts are the points {{{O(0,0)}}} and {{{B(0,-1)}}} .
The distance between the two intercepts of the circle with the x-axis is
{{{0-(-1)=highlight(1)}}} .
 
{{{drawing(300,300,-1,2,-2,1,grid(0),
red(circle(5/6,-1/2,sqrt(34)/6)),
blue(line(0,0,5/3,0)),
blue(line(0,0,0,-1)),
locate(0.05,0,blue(O)),
locate(1.67,0.15,blue(A)),
locate(0.06,-0.87,blue(B))
)}}}