The problem is about finding the derivative $\frac{dy}{dx}$ for the given family of curves $x^2 - y^2 = b^2$ and then explaining why two families of curves form orthogonal trajectories.

Step 1: Differentiate the given equation implicitly

The given family of curves is: $x^2 - y^2 = b^2$ Since $b^2$ is a constant, we'll implicitly differentiate both sides of the equation with respect to $x$.

$\frac{d}{dx} \left( x^2 - y^2 \right) = \frac{d}{dx}(b^2)$

The derivative of $x^2$ with respect to $x$ is $2x$, and for $-y^2$, applying the chain rule gives $-2y \frac{dy}{dx}$. On the right-hand side, the derivative of the constant $b^2$ is 0.

So, we get: $2x - 2y \frac{dy}{dx} = 0$

Step 2: Solve for $\frac{dy}{dx}$

We can now solve for $\frac{dy}{dx}$: $2x = 2y \frac{dy}{dx}$ $\frac{dy}{dx} = \frac{x}{y}$

So, for the given family of curves $x^2 - y^2 = b^2$, we have $\frac{dy}{dx} = \frac{x}{y}$.

Step 3: Orthogonal trajectories

Two families of curves are said to be orthogonal trajectories if the product of their slopes is equal to $-1$ at their points of intersection.

Now, for the family of curves $\frac{dy}{dx} = \frac{x}{y}$, the orthogonal family must have a slope that satisfies: $\frac{dy}{dx} \cdot \left( \text{slope of orthogonal family} \right) = -1$ Substituting $\frac{dy}{dx} = \frac{x}{y}$ into the equation: $\frac{x}{y} \cdot \left( \text{slope of orthogonal family} \right) = -1$ Thus, the slope of the orthogonal family is $\frac{dy}{dx} = -\frac{y}{x}$.

Step 4: Select the correct answer

Now, based on the explanations and calculations, the orthogonal family has the slope $-\frac{y}{x}$. Since this is consistent with Option A, where the families are orthogonal because the product of their slopes is equal to $-1$, the correct answer is:

A. The families are orthogonal because $\frac{dy}{dx} = \frac{x}{y}$ and $\frac{dy}{dx} = -\frac{y}{x}$.

Would you like more details on any step?

Here are five related questions for further practice:

1. How do you find orthogonal trajectories for other types of curves, such as circles?
2. What is implicit differentiation, and why is it used here?
3. How do you find the equation of the tangent line at a given point for a curve?
4. How can you verify whether two families of curves are orthogonal in general?
5. What are some real-world applications of orthogonal trajectories?

Tip: When working with implicit differentiation, always remember to apply the chain rule when differentiating terms involving $y$.