91Ó°ÊÓ

Orthogonal trajectories are an important concept in differential equations, especially when dealing with families of curves. These trajectories are curves that intersect a given family of curves at right angles (orthogonally). To find these trajectories, we need to determine a new family of curves from the original curves where the new curves intersect the original ones perpendicularly.

In the case of the family of ellipses given in this exercise, finding orthogonal trajectories involves taking the derivative of the ellipse equation and replacing the derivative with its negative reciprocal. This substitution results in a new differential equation representing the orthogonal trajectories. Solving this differential equation provides the equations of curves that cross each of the original ellipses at right angles, thereby forming a new family of curves orthogonal to the family of ellipses.

Implicit differentiation is a crucial tool used when differentiation is needed for equations where the dependent and independent variables are intermingled, and it is difficult to solve for one variable in terms of the other. In the exercise, the equation of the ellipse is not solved for one variable completely in terms of the other (like the usual y as a function of x), so implicit differentiation helps us find the derivative without doing so.

By keeping constant terms unaffected by the differentiation and applying the chain rule, we differentiate both x and y simultaneously. This requires careful application of calculus rules, as shown when differentiating the ellipse equation with respect to x, resulting in an equation that expresses the derivative \( \frac{dy}{dx} \) in terms of both x and y.

A family of ellipses refers to a group of ellipses that share a common characteristic or a set of parameters, such as having the same center or a fixed ratio of axes lengths. In this problem, the ellipses are centered at the origin, have a focus at \((c, 0)\), and share a semimajor axis of length \(2c\).

This common property provides a parameter for the family — the distance from the center to the focus, \(c\), and establishes a relationship between the semi-major and semi-minor axes using the equation \(c^2 = a^2 - b^2\). Understanding this family structure is crucial because it allows us to express the general equation for any ellipse within the family and ultimately helps us in deriving the orthogonal trajectories.

Differentiation with respect to x means that x is treated as the variable of interest while any other variables, such as y in this case, are considered dependent on x. This is standard practice in calculus when dealing with functions where y depends on x, which is what's done with implicit differentiation in this exercise.

When differentiating the equation of the ellipses implicitly with respect to x, any derivatives of y are expressed through \( \frac{dy}{dx} \). This is necessary when the original equation contains y in terms of x without an explicit rule (function) separating them. By differentiating this way, we find relationships between the rates of change (slopes) of x and y for the given family of ellipses.