# JEE-Advanced 2015 (15/40)

**Calculus**Level 4

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 ?

\[\begin{array}{} (1) \, P=y+x \quad \quad \quad \quad & (2) \, P=y-x \\ (3) \, P+Q=1-x+y+y'+(y')^2 & (4) \, P+Q=x+y-y'-(y')^2 \end{array}\]

**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.