Question

Orthogonal Trajectories In Exercises 79 and 80 , verify that the two families of curves are orthogonal, where \(C\) and \(K\) are real numbers. Use a graphing utility to graph the two families for two values of \(C\) and two values of \(K\). $$ x y=C, \quad x^{2}-y^{2}=K $$

Short answer

Yes, the families of curves \(xy=C\) and \(x^{2}-y^{2}=K\) are orthogonal.

Step-by-step solution

Find derivative of the first equation

We start by differentiating both sides of the equation \(xy = C\) with respect to \(x\). This gives \(\frac{dy}{dx} = -\frac{C}{x^{2}}\) where \(\frac{dy}{dx}\) is the gradient of the equation \(xy = C\).

Find derivative of the second equation

Next, we differentiate both sides of the equation \(x^{2} - y^{2} = K\) with respect to \(x\). Using the chain rule on \(y2\), this gives \(2x - 2y \frac{dy}{dx} = 0\), or \(\frac{dy}{dx} = \frac{x}{y}\) where \(\frac{dy}{dx}\) is the gradient of the equation \(x^{2} - y^{2} = K\).

Verify Orthogonality

Two families of curves are orthogonal if the product of their gradients is -1. In this case, we multiply the gradients we found: \(\left(-\frac{C}{x^{2}}\right) * \frac{x}{y}\). This simplifies to \(-\frac{C}{xy}= -C\div C= -1\). Therefore, the curves are orthogonal.

Plot the curves

To get a visual representation of the orthogonality, plot the curves using a graphing utility for different values of \(C\) and \(K\). Unfortunately, this cannot be shown in this format, but is a crucial step in understanding this topic.