Irisan Kerucut - Elips • Part 11: Contoh Soal Persamaan Garis Singgung Elips

Introduction

This tutorial is designed to provide a clear and structured guide on deriving the equations of tangent lines to ellipses. It is based on the content of the video "Irisan Kerucut - Elips • Part 11: Contoh Soal Persamaan Garis Singgung Elips" by Jendela Sains. Understanding how to find tangent lines is crucial in geometry and can be applied in various fields such as engineering, physics, and computer graphics.

Step 1: Understanding the Ellipse Equation

• An ellipse can generally be represented by the equation:
  [ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 ] where:

  • (a) is the semi-major axis.
  • (b) is the semi-minor axis.

• Familiarize yourself with the properties of the ellipse, such as the focal points and vertices, as these will aid in understanding the tangent lines.

Step 2: The Concept of Tangent Lines

• A tangent line touches the ellipse at exactly one point.
• The slope of the tangent line at a given point on the ellipse can be derived from implicit differentiation of the ellipse equation.

Step 3: Deriving the Equation of the Tangent Line

1. Identify the point of tangency ((x_0, y_0)) on the ellipse.

2. Differentiate the ellipse equation:

   Use implicit differentiation

   [ \frac{d}{dx}\left(\frac{x^2}{a^2} + \frac{y^2}{b^2}\right) = 0 ]

   This leads to the derivative

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

   Rearranging gives

   [ \frac{dy}{dx} = -\frac{b^2x}{a^2y} ]

3. Calculate the slope at the point ((x_0, y_0)):

   • Substitute (x_0) and (y_0) into the slope equation.

4. Write the equation of the tangent line:

   Use the point-slope form of a line

   [ y - y_0 = m(x - x_0) ] where (m) is the slope calculated in the previous step.

Step 4: Example Problem

• Consider an ellipse with (a = 5) and (b = 3). Find the equation of the tangent line at the point ((3, 2)).

  Verify that ((3, 2)) satisfies the ellipse equation

  [ \frac{3^2}{5^2} + \frac{2^2}{3^2} = \frac{9}{25} + \frac{4}{9} \text{ (check if equal to 1)} ]

  Calculate the slope

  [ m = -\frac{b^2x_0}{a^2y_0} = -\frac{3^2 \cdot 3}{5^2 \cdot 2} = -\frac{27}{50} ]

  Write the tangent line equation

  [ y - 2 = -\frac{27}{50}(x - 3) ]

Conclusion

In this tutorial, we explored how to derive the equation of tangent lines to ellipses. We covered the fundamental concepts of ellipses, the process of differentiation, and provided a concrete example to illustrate the application of these concepts. To deepen your understanding, consider practicing with different points on various ellipses and exploring the implications of tangent lines in real-world scenarios.