Question

Let the equation of the required circle be ( (x-h)^{2}+(y-k)^{2}=r^{2} )
since the circle passes through points (2,3) and (-1,1) ,
( (2-h)^{2}+(3-k)^{2}=r^{2} ldots(1) )
( (-1-h)^{2}+(1-k)^{2}=r^{2} ldots(2) )
since the centre ( (h, k) ) of the circle lies on line ( x-3 y-11=0 )
( h-3 k=11 ldots(3) )
From equations (1) and (2), we obtain
( (2-h)^{2}+(3-k)^{2}=(-1-h)^{2}+(1-k)^{2} )
( 4-4 h+h^{2}+9-6 k+k^{2}=1+2 h+h^{2}+1-2 k+k^{2} )
( 4-4 h+9-6 k=1+2 h+1-2 k )
( 6 h+4 k=11 ldots(4) )
On solving equations (3) and (4), we obtain ( h=frac{7}{2} ) and ( k=frac{-5}{2} )
On substituting the values of ( h ) and ( k ) in equation ( (1), ) we obtain
( left(2-frac{7}{2}right)^{2}+left(3+frac{5}{2}right)^{2}=r^{2} )
( Rightarrowleft(frac{4-7}{2}right)^{2}+left(frac{6+5}{2}right)^{2}=r^{2} )
( Rightarrowleft(frac{-3}{2}right)^{2}+left(frac{11}{2}right)^{2}=r^{2} )
( Rightarrow frac{9}{4}+frac{121}{4}=r^{2} )
( Rightarrow frac{130}{4}=r^{2} )

# the guation of the circle passing through the points (2,3) and (-1.1) and

Solution