Find the orthogonal trajectories of family of curves `x^2+y^2=c x`

Answer

`x^(2)+y^(2)=k^(2)`

Explanation

`x^(2)+y^(2)=cx`…………..(1)
Differentiating w.r.t x, we get `2x+2y(dy)/(dx)=c`……………(2)
Elliminating c between (1) and (2), we get
`2x+2y(dy)/(dx)=(x^(2)+y^(2))/(x)` or `(dy)/(dx) =(y^(2)-x^(2))/(2xy)`
Replacing `(dy)/(dx)by-(dx)/(dy)`, we get `(dy)/(dx)=(2xy)/(x^(2)-y^(2))`
This equation is homogeneous, and its solution gives the orthogonal trajectories as `x^(2)+y^(2)=ky`.