Sol. The equation of circles in the first quadrant which touch the coordinate axes is

(x –a) 2 + (y –a) 2 = a 2 ... (i)

where a is an arbitrary constant. This equation contains one arbitrary constant, so we shall differentiate it once only and we shall get a differential equation of first order.

Differentiating (i) w.r.t. x, we get

2 (x – a) + 2(y – a) = 0

⇒ x – a + (y – a) = 0

⇒ a =

⇒ a = , where p =

Substituting the value of a in (i), we get

+ =

⇒ (xp – py) 2 + (y – x) 2 = (x + py) 2 ⇒ (x – y)

2 p 2 + (x – y) 2 = (x + py) 2 ⇒ (x – y)

2 (p 2 + 1) = (x + py) 2 ⇒ (x – y)

2 =

This is the required differential equation.