The equation of circle is #x^2+y^2+6(x-y)+9=0#.
Now if we have #x=0#, we get #y^2-6y+9=0# or #(y-3)^2=0# i.e. #y=3#. As we have just one point common between #y#-axis and circle, it means it touches #y#-axis at #(0,3)#.
Similarly if we have #y=0#, we get #x^2+6x+9=0# or #(x+3)^2=0# i.e. #x=-3#. As we have just one point common between #x#-axis and circle, it means it touches #x#-axis at #(-3,0)#.
Hence the circle #x^2+y^2+6(x-y)+9=0# touches the two axes.
In a similar way, we can find the points of intersection of circle and line #x-y+4=0# (note that its slope is #-1#), by putting #x=y-4# in equation of circle and we get #(y-4)^2+y^2+6(y-4-y)+9=0# or
#y^2-8y+16+y^2-24+9=0# or #2y^2-8y+1=0# i.e.
#y=(8+-sqrt(64-8))/4=2+-sqrt14/2#
and corresponding values of #x# are #+-sqrt14/2-2# and points are #(sqrt14/2-2,sqrt14/2+2)# and #(-sqrt14/2-2,-sqrt14/2+2)#.
Now to find a circle through #(sqrt14/2-2,sqrt14/2+2)# and #(-sqrt14/2-2,-sqrt14/2+2)# and origin #(0,0)#, we have to find center, which is the intersection of any two perpendicular bisectors.
Observe that mid-point of #(sqrt14/2-2,sqrt14/2+2)# and #(-sqrt14/2-2,-sqrt14/2+2)# is #(-2,2)# and slope of line perpendicular to it would be #-1/1=-1# and hence equation of their perpendicular bisector of points #(sqrt14/2-2,sqrt14/2+2)# and #(-sqrt14/2-2,-sqrt14/2+2)# would be #x+y=0#.
Similarly midpoint of #(sqrt14/2-2,sqrt14/2+2)# and #(0,0)# is #(sqrt14/4-1,sqrt14/4+1)# and slope of line joining them is #(sqrt14/2+2)/(sqrt14/2-2)# and hence slope of their perpendicular bisector is #-(sqrt14/2-2)/(sqrt14/2+2)# and equation is
#y-sqrt14/4-1=-(sqrt14/2-2)/(sqrt14/2+2)(x-sqrt14/4+1)#
or #y(sqrt14/2+2)+x(sqrt14/2-2)=(sqrt14/2-2)(sqrt14/4-1)+(sqrt14/2+2)(sqrt14/4+1)#
or #x(sqrt14-4)+y(sqrt14+4)-15=0#
Solving this and #x+y=0# we get the coordinates of center as #(-15/8,15/8)# and radius as #15/8sqrt2# and equation of circle is
#(x+15/8)^2+(y-15/8)^2=2(15/8)^2#
or #x^2+y^2+15/4x-15/4y=0#
or #4x^2+4y^2+15x-15y=0#
graph{(x^2+y^2+6(x-y)+9)(x-y+4)(4x^2+4y^2+15x-15y)=0 [-12.88, 7.12, -2.72, 7.28]}
