What are Orthogonal Trajectories? | Simple Explanation for Class 12

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What is the Concept of Orthogonal Trajectories?

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

Orthogonal trajectories are two families of curves that intersect each other at right angles (90 degrees) at every single point of intersection. Imagine two different sets of paths, where every path from the first set crosses every path from the second set perfectly perpendicularly.

Simple Example

Quick Example

Think about the lines on a chessboard. The horizontal lines (ranks) form one family of curves. The vertical lines (files) form another family. Where a horizontal line meets a vertical line, they cross at a 90-degree angle. So, the horizontal lines are orthogonal trajectories to the vertical lines.

Worked Example

Step-by-Step

Let's find the orthogonal trajectories for the family of parabolas y^2 = 4ax.

Step 1: Write down the given family of curves: y^2 = 4ax. This is our equation (1).
---Step 2: Differentiate equation (1) with respect to x to find the differential equation of the given family. Using the chain rule, 2y (dy/dx) = 4a.
---Step 3: From equation (1), find 'a' in terms of x and y: a = y^2 / (4x). Substitute this 'a' into the differential equation: 2y (dy/dx) = 4 * (y^2 / (4x)).
---Step 4: Simplify the differential equation: 2y (dy/dx) = y^2 / x. This gives dy/dx = y / (2x). This is the differential equation for the given family.
---Step 5: To find the differential equation of the orthogonal trajectories, replace dy/dx with -1 / (dy/dx). So, -1 / (dy/dx) = y / (2x). This means dy/dx = -2x / y.
---Step 6: Now, solve this new differential equation by separating variables: y dy = -2x dx.
---Step 7: Integrate both sides: integral(y dy) = integral(-2x dx). This gives y^2 / 2 = -x^2 + C, where C is the integration constant.
---Step 8: Rearrange the equation: y^2 / 2 + x^2 = C. Multiplying by 2, we get 2x^2 + y^2 = 2C. Let 2C = K (another constant). So, 2x^2 + y^2 = K.

Answer: The orthogonal trajectories for the family of parabolas y^2 = 4ax are the ellipses represented by the equation 2x^2 + y^2 = K.

Why It Matters

Orthogonal trajectories are super important in engineering and science! They help design things like electric fields in circuits (used in your mobile phone) and understand heat flow in materials (like keeping your laptop cool). In AI/ML, similar concepts help in visualizing data patterns and optimizing algorithms, making smart systems work better.

Common Mistakes

MISTAKE: Forgetting to replace dy/dx with -1/(dy/dx) when finding the differential equation of the orthogonal trajectories. | CORRECTION: Always remember that for orthogonal curves, the product of their slopes at any intersection point is -1. So, if the slope of the original family is m, the slope of the orthogonal trajectory is -1/m.

MISTAKE: Not eliminating the arbitrary constant from the original family's differential equation before finding the orthogonal trajectory's differential equation. | CORRECTION: First, differentiate the given family to get dy/dx. Then, use the original equation to eliminate the arbitrary constant (like 'a' in y^2 = 4ax) from this differential equation. Only then replace dy/dx with -1/(dy/dx).

MISTAKE: Making errors during integration or separation of variables when solving the new differential equation. | CORRECTION: Double-check each step of variable separation and integration carefully. Pay attention to signs and constants of integration.

Practice Questions

Try It Yourself

QUESTION: Find the orthogonal trajectories of the family of straight lines y = mx, where m is an arbitrary constant. | ANSWER: x^2 + y^2 = C (a family of circles centered at the origin)

QUESTION: Determine the orthogonal trajectories for the family of curves x^2 + y^2 = 2ax, where 'a' is an arbitrary constant. | ANSWER: x^2 + y^2 = Cy (a family of circles touching the x-axis at the origin)

QUESTION: Find the orthogonal trajectories of the family of curves y = ce^(-x), where 'c' is an arbitrary constant. | ANSWER: y^2 = 2x + K

MCQ

Quick Quiz

Which of the following is the first step to find the orthogonal trajectories of a given family of curves?

Integrate the given equation.

Differentiate the given equation to find its differential equation and eliminate the constant.

Replace dy/dx with -dx/dy directly.

Substitute a random value for the constant.

The Correct Answer Is:

The first step is always to find the differential equation of the given family of curves, which involves differentiating the equation and eliminating the arbitrary constant. Only after this do you replace dy/dx with -dx/dy.

Real World Connection

In the Real World

Imagine you are an engineer designing a heating system for a large building in Delhi. You need to ensure heat spreads evenly or is directed in specific ways. The lines of heat flow (isotherms) and the lines perpendicular to them (heat flux lines) are orthogonal trajectories. Understanding this helps you place heaters and insulation effectively. Also, in groundwater studies, flow lines and equipotential lines are orthogonal, helping manage water resources.

Key Vocabulary

Key Terms

TRAJECTORY: A path followed by a moving object or a curve in a family. | ORTHOGONAL: Intersecting at right angles (90 degrees). | DIFFERENTIAL EQUATION: An equation involving derivatives of a function. | ARBITRARY CONSTANT: A constant that can take any value, defining a specific curve within a family. | FAMILY OF CURVES: A group of curves that share a common property or equation, differing only by a constant.

What's Next

What to Learn Next

Great job understanding orthogonal trajectories! Next, you can explore 'Exact Differential Equations' and 'Linear Differential Equations'. These build on your knowledge of solving differential equations, which is a key skill for advanced math and science.