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

The general equation of family of circles whose center lies on line y=x is

$\begin{array}{l}{x}^2+y^2-2ax-2ay+c=0\\ Differentiating w.r.t. x, we get\\ 2x+2y{y}^1-2a-2a{y}^1=0\\ \Rightarrow a=\frac{x+y{y}^1}{1+y^1}\\ \text{ Diff. w.r.t }x\\ \left(1+y^1\right)\left[1+y{y}^{11}+{\left(y^1\right)}^2\right]-\left(x+y{y}^1\right)y^{11}=0\\ \Rightarrow (y-x)y^{11}+y^1\left(1+y^1+{\left(y^1\right)}^2\right)+1=0\end{array}$

$\begin{array}{l}\text{ Comparing with G. E. }\\ P=y-x; Q=1+y^1+{\left(y^1\right)}^2\\ P+Q=1-x+y+y^1+{\left(y^1\right)}^2\end{array}$

$\therefore$options (2),(3) are correct