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## Homework Statement

Find the orthogonal trajectories of the given families of curves.

[tex]x^2 + y^2+2Cy=1[/tex]

## Homework Equations

The book has covered homogeneous and separable methods.

## The Attempt at a Solution

To find the orthogonal trajectories, we simply find the curves whose tangents are perpendicular to the tangents of any curves in the original family. Implicit differentiation of the original equation yields

[tex]2x+2y y' + 2C y' = 0[/tex]

Solving for C in the original equation:

[tex]C = \frac{1-x^2-y^2}{2y}[/tex]

Solving for y' and substituting C:

[tex]y' = \frac{-2xy}{1+y^2-x^2}[/tex]

Our solutions are all curves whose tangent lines are the negative reciprocal of this, however this is not a separable or homogeneous equation as far as I can tell, so neither is its reciprocal, and thus I am stuck.

The answer, as given by the book, is

[itex]x^2+y^2-2Cx+1=0[/itex]