Let radius of the circle is a

Now equation of circle in the first quadrant which touches the coordinate axes and having radius a is

(x-a)² + (y-a)² = a² ................1

Now differentiate with respect to x

2(x-a) + 2(y-a)*(dy/dx) = 0

=> (x-a) + (y-a)*(dy/dx) = 0

=> (y-a)*(dy/dx) = -(x-a)

=> y*(dy/dx) - a*(dy/dx) = -x + a

=> y*(dy/dx) + x = a*(dy/dx) + a

=> y*(dy/dx) + x = a*{(dy/dx) + 1}

=> a = {y*(dy/dx) + x}/{(dy/dx) + 1} 

Now put the vaue of a in equation 1 , we get

[x - {y*(dy/dx) + x}/{(dy/dx) + 1}]² + [y - {y*(dy/dx) + x}/{(dy/dx) + 1}]² = [{y*(dy/dx) + x}/{(dy/dx) + 1}]²

After simplify it we get,

(dy/dx)² * (x² - 2xy) + (dy/dx)*(-4x² + 2xy*(dy/dx)) + 4x² + 2xy + y² = 0