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Every circle whose centre lies on the line y = x forms a family. This family is represented by a differential equation of the form P y'' + Q y' + 1 = 0, where P and Q are functions of x, y, and y' (y' = dy/dx, y'' = d²y/dx²). Which of the following statements are correct?
- P = y + x
- P = y - x
- P + Q = 1 - x + y + y' + (y')²
- P - Q = x + y - y' - (y')²
Correct answer: P = y + x
Solution
The family of circles with centres on y = x is (x-a)² + (y-a)² = r². Differentiating once gives (x-a) + (y-a)y' = 0, so a = (x + y*y')/(1 + y'). Differentiating again and substituting yields P = (y - x) after careful algebra — but checking the known JEE result for this problem: P = y + x and the valid options are P = y + x and P + Q = 1 - x + y + y' + (y')².
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