JEE-Advanced 2015 (15/40)

Consider the family of all circles whose centers lie on the straight line $y=x$ . If this family of circles is represented by the differential equation $Py''+Qy'+1=0$ , where $P,Q$ are functions of $x,y$ and $y'$ $\left( \text{ here } y'=\frac{dy}{dx}, y''=\frac{d^2y}{dx^2} \right)$ , then which of the following statements is(are) true ?
$\begin{array}{ll} (1) \, P=y+x \quad \quad \quad \quad & (2) \, P=y-x \\ (3) \, P+Q=1-x+y+y'+(y')^2 & (4) \, P+Q=x+y-y'-(y')^2 \end{array}$
Note :

• Submit your answer as the increasing order of the serial numbers of all the correct options.

• For eg, if your answer is $(1),(2)$ , then submit 12 as the correct answer, if your answer is $(2),(3),(4)$ , then submit 234 as the correct answer.