### Definition and Examples

Let a family of curves be given by the equation

\[g\left( {x,y} \right) = C,\]

where \(C\) is a constant. For the given family of curves, we can draw the orthogonal trajectories, that is another family of curves \(f\left( {x,y} \right) = C\) that cross the given curves at right angles.

For example, the orthogonal trajectory of the family of straight lines defined by the equation \(y = kx,\) where \(k\) is a parameter (the slope of the straight line), is any circle having center at the origin (Figure \(1\)):

\[{x^2} + {y^2} = {R^2},\]

where \(R\) is the radius of the circle.

Similarly, the orthogonal trajectories of the family of ellipses

\[{\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{c^2} – {a^2}}} = 1,\;\;}\kern0pt{\text{where}\;\;}\kern-0.3pt{0 \lt a \lt c,}\]

are confocal hyperbolas satisfying the equation:

\[{\frac{{{x^2}}}{{{b^2}}} – \frac{{{y^2}}}{{{b^2} – {c^2}}} = 1,\;\;}\kern0pt{\text{where}\;\;}\kern-0.3pt{0 \lt c \lt b.}\]

Both families of curves are sketched in Figure \(2.\) Here \(a\) and \(b\) play the role of parameters describing the family of ellipses and hyperbolas, respectively.

### General Method of Finding Orthogonal Trajectories

The common approach for determining orthogonal trajectories is based on solving the partial differential equation:

\[\nabla f\left( {x,y} \right) \cdot \nabla g\left( {x,y} \right) = 0,\]

where the symbol \(\nabla\) means the gradient of the function \(f\left( {x,y} \right)\) or \(g\left( {x,y} \right)\) and the dot means the dot product of the two gradient vectors.

Using the definition of gradient, one can write:

\[{\nabla f\left( {x,y} \right) = \mathbf{grad}\,f\left( {x,y} \right) }={ \left( {\frac{{\partial f}}{{\partial x}},\frac{{\partial f}}{{\partial y}}} \right),}\]

\[{\nabla g\left( {x,y} \right) = \mathbf{grad}\,g\left( {x,y} \right) }={ \left( {\frac{{\partial g}}{{\partial x}},\frac{{\partial g}}{{\partial y}}} \right).}\]

Hence, the partial differential equation is written in the form:

\[

{\nabla f\left( {x,y} \right) \cdot \nabla g\left( {x,y} \right) = 0,\;\;}\Rightarrow

{\left( {\frac{{\partial f}}{{\partial x}},\frac{{\partial f}}{{\partial y}}} \right) \cdot \left( {\frac{{\partial g}}{{\partial x}},\frac{{\partial g}}{{\partial y}}} \right) = 0,\;\;}\Rightarrow

{\frac{{\partial f}}{{\partial x}}\frac{{\partial g}}{{\partial x}} + \frac{{\partial f}}{{\partial y}}\frac{{\partial g}}{{\partial y}} = 0.}

\]

Solving the last PDE, we can determine the equation of the orthogonal trajectories \(f\left( {x,y} \right) = C.\)

### A Practical Algorithm for Constructing Orthogonal Trajectories

Below we describe an easier algorithm for finding orthogonal trajectories \(f\left( {x,y} \right) = C\) of the given family of curves \(g\left( {x,y} \right) = C\) using only ordinary differential equations. The algorithm includes the following steps:

- Construct the differential equation \(G\left( {x,y,y’} \right) = 0\) for the given family of curves \(g\left( {x,y} \right) = C.\) See the web page Differential Equations of Plane Curves about how to do this.
- Replace \(y’\) with \(\left( { – \large\frac{1}{{y’}}\normalsize} \right)\) in this differential equation. As a result, we obtain the differential equation of the orthogonal trajectories.
- Solve the new differential equation to determine the algebraic equation of the family of orthogonal trajectories \(f\left( {x,y} \right) = C.\)

## Solved Problems

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