Consider the family of all circles whose centers lie on the straight line y = x . If this family of circles…

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′ here y′= dydx, y" = d2ydx2 , then which of the following statements is (are) true ?
  1. P=y+x
  2. P=y-x
  3. P+Q=1-x+y+y′+y′2
  4. P-Q=x+y-y′-y′2

Solution

Let required circle be
x2+y2+2gx+y+c=0
Differentiation
2x+2yy′+2g1+y′=0
⇒x+yy′+g1+y′=0 ...(i)
Again differentiation
1+yy′+y2+gy′=0 ...(ii)
Putting value of g in equation (ii),
1+yy′+y′2-x+yy′1+y′y′′=0
So, P=y-x, Q=1+y′+y′2
i.e., P+Q=1-x+y+y′y′2

Asked in: JEE Advanced 2015 (Paper 1)

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