The circle equation is

#C->(x-x_c)^2+(y-y_c)^2=r^2#.

the straight #y = 3/7x+1# can be written as

#-3x+7(y-1)=0# or #(p-p_0).vec v=0# where

#p = (x,y), p_0=(0,1)# and #vec v = (-3,7)#

The parametric representation for the straight line

is given by #p = p_0 + lambda vec v^T# where #vec v^T# is the vector with components #(7,3)# orthogonal to #vec v#. The circle center given by #p_c=(x_c,y_c)# is equidistant from #p_1=(2,1)# and #p_2 = (3,5)#

So we can equate

#norm(p_c-p_1) = norm (p_c-p_2)# but #p_c = p_0+lambda_c vec v^T# so we can state:

#(p_0+lambda_c vec v^T-p_1).(p_0+lambda_c vec v^T-p_1)=(p_0+lambda_c vec v^T-p_1).(p_0+lambda_c vec v^T-p_1)#.

Developing and grouping

#p_1.p_1-2p_0.p_1-2lambda_c vec v^T.p_1 = p_2.p_2-2p_0.p_2-2lambda_c vec v^T.p_2#

or

#p_2.p_2-p_1.p_1-2p_0.(p_2-p_1)-2lambda_c vec v^T.(p_2-p_1)=0#

and finally

#lambda_c = (p_2.p_2-p_1.p_1-2p_0.(p_2-p_1))/(2 vec v^T.(p_2-p_1))#

Substituting values we obtain #lambda_c = 21/38 # then #p_c = (147/38,101/38)#