Let's analyze the two families of curves given:

1. The first curve: $x^2 + y^2 - r^2 = 0$

   • This is the equation of a circle with radius $r$ centered at the origin, i.e., $x^2 + y^2 = r^2$.

2. The second curve: $y - mx = 0$

   • This is the equation of a straight line passing through the origin with slope $m$, i.e., $y = mx$.

Orthogonal Trajectories:

Two families of curves are orthogonal if at every point of intersection, the tangents to the curves are perpendicular. The condition for this is that the product of the slopes of the tangents to the curves at the intersection point must be equal to $-1$.

• For the circle $x^2 + y^2 = r^2$, the slope of the tangent at any point can be found by differentiating implicitly: $2x + 2y \frac{dy}{dx} = 0 \quad \Rightarrow \quad \frac{dy}{dx} = -\frac{x}{y}$ Thus, the slope of the tangent to the circle at any point is $-\frac{x}{y}$.

• For the line $y = mx$, the slope of the tangent is simply $m$.

For the curves to be orthogonal, the product of the slopes of the tangents must be $-1$: $\left( -\frac{x}{y} \right) \times m = -1$ Simplifying this: $\frac{x}{y} \times m = 1 \quad \Rightarrow \quad \frac{x}{y} = \frac{1}{m}$ This condition is satisfied when the point of intersection lies on both the circle and the line, confirming that the two families are indeed orthogonal at the intersection points.

Conclusion:

Since the product of the slopes of the tangents to both curves is $-1$, the two families of curves are orthogonal trajectories.

The correct answer is True.

Would you like more detailed explanations, or do you have any other questions?

Here are some follow-up questions:

1. What is the general method to determine orthogonality for curves?
2. Can you provide an example of two curves that are not orthogonal?
3. How does the geometry of a circle relate to its tangent slopes?
4. What role does implicit differentiation play in finding the tangent slope?
5. What are some common examples of orthogonal families of curves?

Tip: When analyzing orthogonal trajectories, always check if the slopes at the intersection points multiply to $-1$.