JEE-Advanced 2015 (15/40)

Calculus Level 4

Consider the family of all circles whose centers lie on the straight line y=xy=x. If this family of circles is represented by the differential equation Py+Qy+1=0Py''+Qy'+1=0, where P,QP,Q are functions of x,yx,y and yy' ( here y=dydx,y=d2ydx2) \left( \text{ here } y'=\frac{dy}{dx}, y''=\frac{d^2y}{dx^2} \right) , then which of the following statements is(are) true ?
(1)P=y+x(2)P=yx(3)P+Q=1x+y+y+(y)2(4)P+Q=x+yy(y)2\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)(1),(2), then submit 12 as the correct answer, if your answer is (2),(3),(4)(2),(3),(4), then submit 234 as the correct answer.

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