The given family of conics is described by the equation:

$\frac{x^2}{a^2 + \mu} + \frac{y^2}{b^2 + \mu} = 1,$

where $\mu$ is a parameter. We are tasked with finding the orthogonal trajectories of this family of conics. Here's the process for solving it:

Step 1: Implicit differentiation

Differentiate the given equation with respect to $x$, treating $\mu$ as constant since it is a parameter:

$\frac{\partial}{\partial x} \left( \frac{x^2}{a^2 + \mu} + \frac{y^2}{b^2 + \mu} \right) = \frac{\partial}{\partial x} 1.$

This yields:

$\frac{2x}{a^2 + \mu} + \frac{2y}{b^2 + \mu} \frac{dy}{dx} = 0.$

Simplifying:

$\frac{x}{a^2 + \mu} + \frac{y}{b^2 + \mu} \frac{dy}{dx} = 0.$

Solve for $\frac{dy}{dx}$:

$\frac{dy}{dx} = - \frac{x(b^2 + \mu)}{y(a^2 + \mu)}.$

Step 2: Orthogonal trajectories

For orthogonal trajectories, the slopes of the trajectories must satisfy the condition:

$\frac{dy}{dx} \cdot \frac{dy_\perp}{dx} = -1,$

where $\frac{dy_\perp}{dx}$ is the slope of the orthogonal trajectories. Thus, the slope of the orthogonal trajectories is given by:

$\frac{dy_\perp}{dx} = \frac{y(a^2 + \mu)}{x(b^2 + \mu)}.$

Step 3: Solving the differential equation

Now, we have the differential equation for the orthogonal trajectories:

$\frac{dy}{dx} = \frac{y(a^2 + \mu)}{x(b^2 + \mu)}.$

This is a separable differential equation. Rearranging the terms:

$\frac{dy}{y} = \frac{(a^2 + \mu)}{(b^2 + \mu)} \frac{dx}{x}.$

Integrating both sides:

$\ln |y| = \frac{a^2 + \mu}{b^2 + \mu} \ln |x| + C,$

where $C$ is the constant of integration.

Exponentiating both sides:

$|y| = e^C |x|^{\frac{a^2 + \mu}{b^2 + \mu}}.$

Thus, the equation of the orthogonal trajectories is:

$y = C' x^{\frac{a^2 + \mu}{b^2 + \mu}},$

where $C' = e^C$ is a new constant.

Conclusion:

The orthogonal trajectories of the given family of conics are described by the equation:

$y = C' x^{\frac{a^2 + \mu}{b^2 + \mu}}.$

This equation represents the family of curves orthogonal to the original family of conics.

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Would you like further details or have any questions on this solution?

Here are five related questions to expand on this topic:

1. How can we confirm that the two families of curves are orthogonal geometrically?
2. How would the process change if the parameter $\mu$ were not constant but a function of $x$ or $y$?
3. How do orthogonal trajectories apply to electrostatic or fluid flow problems?
4. What happens if $a = b$ in the given family of conics?
5. Can we generalize this method to three-dimensional surfaces?

Tip: Orthogonal trajectories are a powerful tool for analyzing families of curves, particularly in physics, where they describe field lines in electrostatics or fluid dynamics.