General equation of circle with centre on x-axis: ⇒ x2+y2 +2fx + C = 0 where centre (-f, 0) and radius = √f2−C this circle passes from (2, 3) so, (2)2+(3)2 + 2f(2) + C = 0 ⇒4f + C =-13 ⇒C=-13-4f Equation of circle = x2+y2 + 2fx - 4f - 13 = 0 ...(i) Differentiate this equation with respect to x. 2x + 2y
dy
dx
+ 2f = 0 ⇒ 𝑓 = −𝑥 − 𝑦
dy
dx
put the value of f in equation of circle. ⇒ x2+y2+2x(−x−y
dy
dx
) - 4(−x−y
dy
dx
) - 13 = 0 ⇒ y2−x2 - 2xy y' + 4yy' + 4x -13 = 0 Thus, the above equation is the differential equation for the given circle.