Understanding Orthogonal Trajectories

Orthogonal trajectories are a family of curves that intersect every curve in a given family of curves at a right angle (90 degrees). To find the orthogonal trajectories of a given family of curves, we follow a standard procedure involving differential equations.

Finding the Differential Equation of the Given Family

The given family of hyperbolas is represented by the equation:

\( x^2 - y^2 = c \)

where \(c\) is an arbitrary constant. To find the differential equation for this family, we differentiate the equation implicitly with respect to \(x\):

\( \frac{d}{dx}(x^2) - \frac{d}{dx}(y^2) = \frac{d}{dx}(c) \)

Applying differentiation rules:

\( 2x - 2y \frac{dy}{dx} = 0 \)

Now, we solve for \( \frac{dy}{dx} \) to find the slope of the tangent line to any curve in the family at a point \( (x, y) \):

\( 2y \frac{dy}{dx} = 2x \)

\( \frac{dy}{dx} = \frac{2x}{2y} \)

\( \frac{dy}{dx} = \frac{x}{y} \)

This is the differential equation representing the given family of hyperbolas \( x^2 - y^2 = c \).

Determining the Differential Equation for Orthogonal Trajectories

For two curves to intersect orthogonally, the product of their slopes at the point of intersection must be -1. If \( \frac{dy}{dx} \) is the slope of a curve from the original family, the slope of its orthogonal trajectory at the same point must be \( -\frac{1}{\frac{dy}{dx}} \) or \( -\frac{dx}{dy} \). The differential equation for the orthogonal trajectories is obtained by replacing \( \frac{dy}{dx} \) with \( -\frac{dx}{dy} \) in the differential equation of the original family.

The differential equation for the original family is \( \frac{dy}{dx} = \frac{x}{y} \). Replacing \( \frac{dy}{dx} \) with \( -\frac{dx}{dy} \), we get:

\( -\frac{dx}{dy} = \frac{x}{y} \)

Rearranging this to get \( \frac{dy}{dx} \) for the orthogonal trajectories:

\( \frac{dy}{dx} = -\frac{y}{x} \)

This is the differential equation that represents the family of orthogonal trajectories.

Solving the Differential Equation for Orthogonal Trajectories

The differential equation for the orthogonal trajectories is \( \frac{dy}{dx} = -\frac{y}{x} \). This is a separable differential equation. We can separate the variables \(y\) and \(x\) and integrate:

\( \frac{dy}{y} = -\frac{dx}{x} \)

Integrate both sides:

\( \int \frac{dy}{y} = \int -\frac{dx}{x} \)

\( \ln|y| = -\ln|x| + C' \)

where \(C'\) is the constant of integration. We can rewrite the constant \(C'\) as \( \ln|C| \) for a positive constant \(C\), which simplifies the equation:

\( \ln|y| = -\ln|x| + \ln|C| \)

Using logarithm properties (\( \ln a - \ln b = \ln \frac{a}{b} \) and \( \ln a + \ln b = \ln(ab) \)):

\( \ln|y| + \ln|x| = \ln|C| \)

\( \ln|xy| = \ln|C| \)

Exponentiating both sides (taking antilog):

\( |xy| = |C| \)

Since \(C\) is an arbitrary constant, \( \pm C \) can be represented by a new arbitrary constant, which we can still call \(c\).

\( xy = c \)

This is the equation of the orthogonal trajectories of the hyperbolas \( x^2 - y^2 = c \).

Comparing with Options

Let's compare our result \( xy = c \) with the given options:

1. x² + y² = c
2. x + y = c
3. xy = c
4. x² + y² + 2x = c

Our derived equation \( xy = c \) matches Option 3.

Revision Table: Steps to Find Orthogonal Trajectories

Additional Information on Orthogonal Trajectories and Differential Equations

The concept of orthogonal trajectories is important in various fields, including physics (e.g., level curves of temperature and heat flow lines are orthogonal trajectories) and mathematics (e.g., understanding families of curves). The process relies heavily on the ability to formulate and solve differential equations.

• Family of Curves: A set of curves whose equations differ only by the value of one or more parameters (like \(c\) in \( x^2 - y^2 = c \)). Each specific value of the parameter gives a unique curve within the family.
• Differential Equation: An equation that relates a function with its derivatives. The order of the differential equation is the order of the highest derivative involved. The differential equation \( \frac{dy}{dx} = f(x, y) \) gives the slope of the tangent to the curve at any point \((x, y)\).
• Separable Differential Equations: A type of first-order differential equation that can be written in the form \( \frac{dy}{dx} = f(x)g(y) \). These can be solved by separating variables: \( \frac{dy}{g(y)} = f(x) dx \) and integrating both sides. The differential equation \( \frac{dy}{dx} = -\frac{y}{x} \) we solved is separable.

Finding orthogonal trajectories is a classic application of first-order differential equations.