The equation of the circle passing through (0, 0) and belonging to the system of circles of which (3, 1) and (-1, 5) are limiting points, is

Circles having( 3, 1) and (-1, 5) as limiting pointsare S1≡(x−3)2+(y−1)2=0 and S2≡(x+1)2+(y−5)2=0The equation of the family of circles isS1+λS1−S2=0⇒ (x−3)2+(y−1)2+λ(−8x+8y−16)=0                              ... (i)It passes through (0, 0).∴ 10−16λ=0⇒λ=58Substituting the value of λ in (i), we getx2+y2−11x+3y=0 as the equation of the required circle.