Orthogonal Trajectories and Parameters What's the difference between constants and parameters in terms of finding orthogonal trajectories to a given family of curves? 
I understand that for a curve like
$x^2 + y^2 + 2gx + c = 0$, 
the process of finding an orthogonal trajectory involves finding the gradient $dy/dx$ in terms of y and x only. We can then use this expression to calculate the orthogonal trajectory's gradient (the negative reciprocal of $dy/dx$) and solve it to get a function of y in terms of x.
My textbook says g is a parameter but says nothing about c. Does this mean g will appear in the expression for the orthogonal trajectory? If so, is this because a parameter in the equation of a family of curves affects the shape of the orthogonal trajectory and hence must appear in the expression for the latter?
I've worked out the orthogonal trajectory to be 
$y = k(g + x)$ using this method. Please let me know if and where I'm making a mistake. 
 A: You are doing it right.
$$x^2+y^2+2gx+c=0$$
$$x^2+2gx+(g)^2-g^2+y^2+c=0$$
$$(x+g)^2+y^2=g^2-c$$
There you go the equation of a typical circle. centered at $(-g,0)$
The radius of the circle is then $\sqrt{g^2+c}$
Finding orthogonal trajectory, we have three steps.
Step 1: Differentiating
$$2(x+g)+2y\frac{dy}{dx}=0$$
$$\frac{dy}{dx}=-\frac{x+g}{y}$$
Step 2: Taking negative reciprocal of the differential equation
$$\frac{dy}{dx}=\frac{y}{x+g}$$
Step 3: Solving the differential equation
$$\frac{dy}{dx}=\frac{y}{x+g}$$
$$\frac{dy}{y}=\frac{dx}{x+g}$$
$$\int \frac{dy}{y}=\int \frac{dx}{x+g} $$
$$\ln|y|=\ln|x+g|+k$$
$$y=k(x+g)$$
This is an equation of a straight line.
It will look something like this,

Notice that the linear equations always has a root at $x=-g$ which is also the center of the circle equation. You are free to choose which center $g$.
A: The equation $x^2 + y^2 + 2gx + c = 0$ represents a family of circles whose centers are $(-g;\;0)$ and radii are $g^2-c$ thus the orthogonal curves are infinite, too. Namely they are all the straight lines passing through the center of the circle and their equation is $y=k (g+x)$

