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find the orthogonal trajectories of the family of curves x^2+y^2=2gx where g is arbitary constant

find the orthogonal trajectories of the family of curves x^2+y^2=2gx
where g is arbitary constant

Grade:12

1 Answers

Vikas TU
14149 Points
4 years ago
Dear student 
x^2+y^2=2gx .......(1)
Differentiate wrt x 
2x + 2ydy/dx = 2g 
Find the value of g from here and put it to equation 1 
x^2+y^2= x (x + ydy/dx) 
Replace dy/dx with -dx /dy 
(y^2 - x^2 )/ xy = -dx /dy 
Put x = ty 
dx /dy = dt/dy *y + t 
put in place of dx /dy 
inegral (2t/(1+t^2) dt = -(integral (dy/y)) 
ln(1+t^2) = -lny + lnK 
Final ans is 
x^2 + y^2 = ky 
 

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