Geometri Analitik Fase F | Persamaan Lingkaran Bagian 2 - Kedudukan Titik Terhadap Lingkaran

Introduction

This tutorial focuses on determining the position of a point relative to a circle using analytic geometry. It is a continuation of the previous discussion on circle equations and is intended for high school students, math teachers, and anyone interested in deepening their understanding of analytic geometry.

Step 1: Understanding the Circle Equation

To analyze the position of a point concerning a circle, you need to know the equation of the circle, which is generally expressed as:

• (x - h)² + (y - k)² = r²

Where:

• (h, k) is the center of the circle.
• r is the radius of the circle.

Practical Tips

• Ensure you can identify the center and radius from the equation.
• If the equation is given in a different format, convert it to the standard form.

Step 2: Determining the Position of a Point

To determine whether a point (x₀, y₀) is inside, outside, or on the circle, follow these steps:

1. Calculate the distance from the center to the point using the formula:

   • Distance = √((x₀ - h)² + (y₀ - k)²)

2. Compare the distance to the radius:

   • If Distance < r: The point is inside the circle.
   • If Distance = r: The point is on the circle.
   • If Distance > r: The point is outside the circle.

Example

• Given the circle (x - 2)² + (y - 3)² = 16 (where h=2, k=3, r=4) and the point (3, 4):
  • Calculate the distance:
    • Distance = √((3 - 2)² + (4 - 3)²) = √(1 + 1) = √2
  • Compare √2 to the radius 4:
    • Since √2 < 4, the point (3, 4) is inside the circle.

Step 3: Solving Practice Problems

To reinforce your understanding, practice with the following steps:

1. Find the equation of the circle given its center and radius.
2. Choose various points and determine their positions relative to the circle.
3. Check your answers against known solutions or through peer discussion.

Common Pitfalls

• Miscalculating the distance formula.
• Confusing the radius with the diameter.
• Not converting the circle equation to standard form.

Conclusion

Understanding how to determine the position of a point relative to a circle is a fundamental skill in analytic geometry. By practicing with different points and circle equations, you can reinforce your learning and prepare for exams. For further study, consider exploring related topics such as line and circle relations or systems of equations involving circles.