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CGP EDU Academic Team
Published on: August 13, 2026
Find the orthogonal trajectories of the circles x 2 + y 2 –ay = 0, where a is a parameter.
Text Solution
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Sol. We have,
x 2 + y 2 –ay = 0... (i)
⇒ 2x + 2y 
– a 
= 0
⇒ a = 
Substituting the value of a in (i), we obtain
x 2 + y 2 –2y 
= 0
⇒ (x 2 –y 2 ) 
–2xy = 0
This is the differential equation of the family of circles given in (i). The differential equation representing the orthogonal trajectories is
–(x 2 –y 2 ) 
–2xy = 0
⇒ 
= – 
⇒ 2xy dy – y 2 dx = –x 2 dx
⇒ 
= –dx
⇒ d 
= –dx
⇒ 
= –x + C [On integration]
⇒ y 2 + x 2 = Cx
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