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Orthogonal trajectories of the family of curves (x - 1)2 + y2 + 2ax = 0 are the solution of the differential equation

  • a)

  • b)

  • c)
    x2 - y2 - 1 - 2xydy/dx = 0(d)

  • d)

Correct answer is option 'A'. Can you explain this answer?
Verified Answer
Orthogonal trajectories of the family of curves (x - 1)2 + y2 + 2ax = ...
we have (x - 1)2 + y2 + 2ax = 0 ...(i)
Differentiating w.r.t. x, we get 
2(x - 1) + 2y y' + 2a = 0
implies 

Putting expression of a in eq. (i), we get
(x - 1)2 + y2 - 2x((x - 1) + yy') = 0
implies x2 - 2x + 1 + y2 - 2x2 + 2x + 2xyy' = 0 

Orthogonal trajectories are given by
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Orthogonal trajectories of the family of curves (x - 1)2 + y2 + 2ax = ...


By putting the value of a from equation (ii) in equation (i), we get.


It ’s orthogonal trajectory will be D.E.

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