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.

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