As the circle passes through points #(5,7)# and #(3,2)#, its center is equidistant from these two points and hence lies on perpendicular bisector of points #(5,7)# and #(3,2)#, whose equation is given by
#(x-5)^2+(y-7)^2=(x-3)^2+(y-2)^2#
or #x^2-10x+25+y^2-14y+49=x^2-6x+9+y^2-4y+4#
or #-10x+25-14y+49+6x-9+4y-4=0#
or #-4x-10y+61=0# or #4x+10y=61#......................(A)
As center also lies on #y=2/3x+7#, putting this in (A)
#4x+10(2/3x+7)=61# or #12x+20x+210=183#
or #32x=183-210=-27# and #x=-27/32# and
#y=2/3xx(-27/32)+7=-9/16+7=103/16#
Hence, center is #(-27/32,103/16)# and radius squared is #(-27/32-3)^2+(103/16-2)^2#
And equation of circle is
#(x+27/32)^2+(y-103/16)^2=(-27/32-3)^2+(103/16-2)^2#
or #x^2+27/16x+(27/32)^2+y^2-103/8y+(103/16)^2=(123/32)^2+(71/16)^2#
Multiplying by #32^2=1024#, we get
#1024x^2+27xx64x+729+1024y^2-103xx128y+206^2=123^2+142^2#
or #1024x^2+1728x+729+1024y^2-13184y+42436=15129+20164#
or #1024x^2+1024y^2+1728x-13184y+7872=0# and dividing by #8#
#128x^2+128y^2+216x-1648y+984=0# and dividing again by #8#
#16x^2+16y^2+27x-206y+123=0#
