Find the orthogonal trajectories of the family of conics

x
2
a
2 + μ
+
y
2
b
2 + μ
= 1,

where μ is a parameter.

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.

---

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.

Find the orthogonal trajectories of the family of conics: x^2 / (a^2 + μ) + y^2 / (b^2 + μ) = 1, where μ is a parameter.

Discover how to find the orthogonal trajectories of the family of conics described by x^2 / (a^2 + μ) + y^2 / (b^2 + μ) = 1. This solution involves differential equations, implicit differentiation, and key geometric concepts. Suitable for undergraduate-level mathematics.

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