I tried solving for the orthogonal trajectories of the following three families:

- $\displaystyle y = mx$
- $\displaystyle xy = c^2$
- $\displaystyle x^\frac{3}{2} + y^\frac{3}{2} = a^\frac{3}{2}$

here are my solutions:

and my answers for the respective problems were:

- $\displaystyle y^2 + x^2 = 2c$
- $\displaystyle y^2 - x^2 = 2c$
- $\displaystyle \sqrt{y} - \sqrt{x} = \frac{c}{2}$

But the answers from the book are, respectively:

- $\displaystyle y^2 + x^2 = a^2$
- $\displaystyle x^2 - y^2 = a^2$
- $\displaystyle \sqrt{y} - \sqrt{x} = \sqrt{a}$

What am I doing wrong or not understanding or how do I make my answers more consistent with the proper format as per in the book? Especially in the 2nd problem on my left hand side i got $\displaystyle y^2 - x^2$ as opposed to the answer's left hand side which is $\displaystyle x^2 - y^2$ please help me

He is changing the constant 'c' into 'a' on purpose? and if so then how do you know what you're suppose to change 'c' into?