eliminate the arbitrary constant \frac{x^2}{a^2}+\frac{y^2}{b^2}=1

To eliminate the arbitrary constant in the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), we can differentiate the equation with respect to \(x\).

### Differentiation:

Given the equation:
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]

Differentiate both sides with respect to \(x\):
\[
\frac{d}{dx} \left(\frac{x^2}{a^2} + \frac{y^2}{b^2}\right) = \frac{d}{dx} (1)
\]

The right side is zero since the derivative of a constant is zero:
\[
\frac{d}{dx} \left(\frac{x^2}{a^2}\right) + \frac{d}{dx} \left(\frac{y^2}{b^2}\right) = 0
\]

### Differentiation of each term:
1. \(\frac{d}{dx} \left(\frac{x^2}{a^2}\right) = \frac{2x}{a^2}\)
2. \(\frac{d}{dx} \left(\frac{y^2}{b^2}\right)\) requires the chain rule:
   \[
   \frac{d}{dx} \left(\frac{y^2}{b^2}\right) = \frac{2y}{b^2} \cdot \frac{dy}{dx}
   \]

### Substituting into the equation:
\[
\frac{2x}{a^2} + \frac{2y}{b^2} \cdot \frac{dy}{dx} = 0
\]

### Simplifying:
\[
\frac{x}{a^2} + \frac{y}{b^2} \cdot \frac{dy}{dx} = 0
\]

Solve for \(\frac{dy}{dx}\):
\[
\frac{dy}{dx} = -\frac{b^2}{a^2} \cdot \frac{x}{y}
\]

This differential equation no longer contains the arbitrary constants \(a\) and \(b\).

### Summary:
The differential equation \(\frac{dy}{dx} = -\frac{b^2}{a^2} \cdot \frac{x}{y}\) is the result of eliminating the arbitrary constants from the original equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

Would you like a deeper explanation, or do you have any specific questions?

### Relative Questions:
1. What is the significance of eliminating arbitrary constants in an equation?
2. How would you solve the differential equation \(\frac{dy}{dx} = -\frac{b^2}{a^2} \cdot \frac{x}{y}\)?
3. What are some common techniques for eliminating arbitrary constants?
4. Can arbitrary constants always be eliminated from any equation?
5. How do the parameters \(a\) and \(b\) affect the geometry of the original ellipse equation?

### Tip:
When differentiating implicitly, always apply the chain rule carefully, especially when dealing with functions of multiple variables.

Learn how to eliminate arbitrary constants in the ellipse equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) using differentiation. Explore the differential equation \(\frac{dy}{dx} = -\frac{b^2}{a^2} \cdot \frac{x}{y}\) derived from this process. Suitable for advanced learners interested in differential equations and ellipse geometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation