Question

find the orthogonal traject of the family of curves rncosn0 = ​

Answer 1

Step-by-step explanation:

The orthogonal trajectories are the curves that are perpendicular to the family everywhere. In other words, the orthogonal trajectories are another family of curves in which each curve is perpendicular to the curves in original family. In the example below we’ll show how to use calculus to find the orthogonal trajectories, but for now we’ll give away the answer so that we can sketch the family of orthogonal trajectories and see that they are perpendicular to the original family.

orthogonal trajectories to the family of curves.png

The curves in blue are from the original family y=kxy=kx. The curves in green are

x^2+y^2=1x

2

+y

2

=1

x^2+y^2=2x

2

+y

2

=2

x^2+y^2=3x

2

+y

2

=3

x^2+y^2=4x

2

+y

2

=4

which are four of their orthogonal trajectories, part of their whole family of orthogonal trajectories given by x^2+y^2=Cx

2

+y

2

=C. Notice how each green circle is perpendicular to every blue line, wherever a green and blue curve intersect one another.

Now that we have an idea of what we’re trying to find, let’s try an example where we use calculus to show that the orthogonal trajectories are given by x^2+y^2=Cx

2

+y

2

=C.