JEE Advanced | 2015CircleHard
Question
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′ = 
), then which of the following statements is (are) true ?
), then which of the following statements is (are) true ?Options
A.P = y + x
B.P = y − x
C.P + Q = 1 − x + y + y′ + (y′)2
D.P − Q = x + y − y′ − (y′)2
Solution
Let Circle
x2 + y2 - 2ax - 2ay + c = 0
On differentiation
2x + 2yy′ - 2a - 2ay′ = 0
⇒ x + yy′ - a(1 + y′) = 0
⇒ a =
again differentiation
⇒ 1 + y′((y′)2 + y′ + 1) + y″(y - x) = 0
∴ P = y - x
Q = 1 + y′ + (y′)2
x2 + y2 - 2ax - 2ay + c = 0
On differentiation
2x + 2yy′ - 2a - 2ay′ = 0
⇒ x + yy′ - a(1 + y′) = 0
⇒ a =
again differentiation
⇒ 1 + y′((y′)2 + y′ + 1) + y″(y - x) = 0
∴ P = y - x
Q = 1 + y′ + (y′)2
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