Find the orthogonal trajectory.

*y* = *ke*−2*x*

To find the orthogonal trajectory of the given family of curves \( y = k e^{-2x} \), we follow these steps:

### Step 1: Differentiate the given family of curves
We start by finding the derivative of \( y = k e^{-2x} \) with respect to \( x \), implicitly treating \( k \) as a constant.

\[
\frac{dy}{dx} = \frac{d}{dx}(k e^{-2x}) = k(-2)e^{-2x} = -2y
\]
Thus, the slope of the tangents to the curves is:
\[
\frac{dy}{dx} = -2y
\]

### Step 2: Find the slope of the orthogonal trajectory
The orthogonal trajectories must have slopes that are negative reciprocals of the original slope. If the original slope is \( \frac{dy}{dx} = -2y \), then for the orthogonal trajectory, the slope is:
\[
\frac{dy}{dx}_{\text{orthogonal}} = \frac{1}{2y}
\]

### Step 3: Set up the differential equation
Now, we set up a differential equation for the orthogonal trajectories:
\[
\frac{dy}{dx} = \frac{1}{2y}
\]

### Step 4: Separate the variables
We can solve this differential equation by separating the variables:
\[
2y \, dy = dx
\]

### Step 5: Integrate both sides
Now, integrate both sides:
\[
\int 2y \, dy = \int dx
\]
\[
y^2 = x + C
\]
Thus, the equation of the orthogonal trajectories is:
\[
y^2 = x + C
\]
where \( C \) is the constant of integration.

### Final Answer:
The orthogonal trajectories to the family of curves \( y = k e^{-2x} \) are given by the family of curves:
\[
y^2 = x + C
\]

---

Would you like further details on any step or additional clarifications?

### Related Questions:
1. How do you find the orthogonal trajectory for a family of parabolas?
2. What is the geometric meaning of orthogonal trajectories?
3. Can orthogonal trajectories always be expressed in a simple form like this?
4. How does the sign of the reciprocal slope influence the trajectory shape?
5. What happens when orthogonal trajectories meet at singular points?

**Tip**: Always check if a family of curves can be written explicitly before starting with differentiation—it simplifies the process!

Find the orthogonal trajectory for the family of curves y = k e^(-2x).

Discover how to find the orthogonal trajectory for the family of curves y = k e^(-2x) with a step-by-step solution. This solution involves differentiating the given family of curves, finding the reciprocal slope, and solving the resulting differential equation. Suitable for Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation