General equation of a circle can be given as x2+y2 + 2gx + 2fy + c = 0 Since it is given that this circle passes through origin and centre lies on x - axis ∴ Centre of circle is (- g , 0) so, the equation of a circle passing through origin and centre on x - axis is : x2+y2 + 2gx = 0 ... (1) ⇒ g = -
x2+y2
2x
Differentiating (1) wrt x 2x + 2y
dy
dx
−
x2+y2
x
= 0 ⇒ 2x+2y
dy
dx
=
x2+y2
x
⇒ 2x2+2xy
dy
dx
= x2+y2 Re - arranging this , we get ⇒ (y2−x2)dx - 2xydy = 0 which is the required differential equation Hence, option B.