Consider the circle x2+y2−10x−6y+30=0. Let O be the center of the circle and tangents at A(7 , 3) andB(5, 1) me et at C. Let S = 0 represents the family of circles passing through A and B. Then

The coordinates of O are (5, 3) and the radius is 2.The equation of tangent at A(7 ,3) is7x+3y−5(x+7)−3(y+3)+30=0i.e., 2x - 14 = 0i.e., x = 7The equation of tangent at B(5, 1) is5x+y−5(x+5)−3(y+1)+30=0i.e., −2y+2=0i.e., y = 1Therefore, the coordinates of C are (7,1). So,area of OACB = 4The equation of AB is x - y = 4 (radical axis).The equation of the smallest circle is(x−7)(x−5)+(y−3)(y−1)=0i.e., x2+y2−12x−4y+38=0