Family of Circles
The given family of circles is represented by the equation x² + y² + 2gx + c = 0, where g is the parameter and c is a constant.

Finding the Orthogonal Trajectories
To find the orthogonal trajectories of the given family of circles, we need to determine the differential equation satisfied by these orthogonal trajectories.

Step 1: Differentiating the Given Equation
Differentiating the equation of the family of circles with respect to x, we get:
2x + 2yy' + 2g = 0

Step 2: Finding the Slope of the Orthogonal Trajectories
The slopes of the orthogonal trajectories are negative reciprocals of the slopes of the family of circles. Therefore, the slope of the orthogonal trajectories is given by:
m_orthogonal = -1/m

Step 3: Expressing the Differential Equation
Using the slope of the orthogonal trajectories, we can express the differential equation as follows:
2x + 2yy' + 2g = -1/m

Step 4: Simplifying the Differential Equation
To simplify the differential equation, we can eliminate the parameter m by expressing it in terms of x and y. Rearranging the equation, we have:
y' = (-x - g)/y

Step 5: Separating Variables
To solve the differential equation, we separate the variables by multiplying both sides by y and dx:
ydy = (-x - g)dx

Step 6: Integrating Both Sides
Integrating both sides of the equation, we have:
∫ydy = -∫(x + g)dx

Simplifying the integrals, we get:
(y^2)/2 = -(x^2)/2 - gx + C

Step 7: Rearranging the Equation
Rearranging the equation, we have:
x^2 + y^2 + 2gx + C = 0

This equation represents the family of circles orthogonal to the given family of circles.

Conclusion
The orthogonal trajectories of the family of circles x² + y² + 2gx + c = 0 are given by the equation x^2 + y^2 + 2gx + C = 0, where C is a constant. These orthogonal trajectories are circles with centers (-g, 0) and radii √(g^2 - C).