Step 1:

The equation is of the form $(x-a)^2 + (y-a)^2 = a^2$-------(1)

On differentiating this with respect to $x$ we get,

$2(x-a)+2(y-a).y'=0$

dividing throughout by 2 we get,

$(x-a) + (y-a).y' = 0x+yy' = a(1+y')$

$a=\large\frac{(x+yy')}{(1+y')}$

Step 2:

Substituting this value of a in equ(1) we get

$[x -\large\frac{ (x+yy')}{(1+y')}]^2 +[y- \large\frac {(x+yy')}{(1+y')}]^2 = \big[\large\frac{x+yy'}{1+y'}\big]^2$

$[x(1+y') - x - yy']^2 + [y(1+y') - x - yy']^2 = [x+yy']^2$

On expanding we get,

$(xy' -yy')^2 + (y - x)^2 = (x+yy')^2$

$(x-y)^2(1 + y')^2 = (x+yy')^2$

This is the required solution.