Consider the family of all circles whose centres lie on the straight line $x+y=0$  and passing through the origin. If this family of circle is represented by the differential equation
  $A{y}^{11}+B{y}^1+1=0$, where A,B are functions of  $x,yand{y}^1(here{y}^1=\frac{dy}{dx},y^{11}=\frac{d^{2}y}{d{x}^2})$, then which of the following statements is(are) true ?

Let the equation of the circle be $x^2+y^2+2gx-2gy=0$
 $2x+2y{y}^1+2g-2g{y}^1=0$
$x+y{y}^1+g-g{y}^1=0$ ____(1)
Again differentiating
$1+y{y}^{11}+{\left(y^1\right)}^2-g{y}^{11}=0$

$g=\frac{1+y{y}^1+{\left(y^1\right)}^2}{y^{11}} sub. in \_\_\_\_\_\left(1\right)$

$$\begin{gathered} x+y{y}^1+\frac{1+{\left(y^1\right)}^2+y{y}^{11}}{y^{11}}-\left(\frac{1+{\left(y^1\right)}^2+y{y}^{11}}{y^{11}}\right)y^1=0 \\ x{y}^{11}+y{y}^1{y}^{11}+1+{\left(y^1\right)}^2+y{y}^{11}-y^1-{\left(y^1\right)}^3-y{y}^1{y}^{11}=0 \\ (x+y)y^{11}-y^1(1-y^1+{\left(y^1\right)}^2)+1=0 \\ A{y}^{11}+B{y}^1+1=0 \\ A=x+y;B=-(1-y^1+{\left(y^1\right)}^2) \\ A-B=1+x+y-y^1+{\left(y^1\right)}^2 \end{gathered}$$