Determine the Equation of a Circle Touching Specific Lines: A Step-by-Step Guide

Understanding how to find the equation of a circle, especially when it touches specific lines, is a vital aspect of both algebra and geometry. In this detailed guide, we will walk you through the process of determining the equation of a circle that touches the lines x0, y4, and y10. This article is designed to cater to learners who are new to the subject and experienced mathematicians alike.

Understanding the Problem

The problem at hand involves a circle that touches the y-axis (x0) and two horizontal lines at y4 and y10. Here's what we know:

The y-axis intersects the circle at a point where x0. The vertical distance between the lines y4 and y10 is 6 units, which is the diameter of the circle. The radius (r) of the circle is therefore 3 units, because the diameter equals 2r. The center of the circle is equidistant from the two horizontal lines and lies along the x-axis.

Finding the Center of the Circle

The center of the circle is located 3 units away from the y-axis (since the radius is 3 units) and 4 units above the horizontal line y4 (since the center is equidistant from y4 and y10).

From the information given, the center is at (3, 7). The distance from the center to the line y4 is 3 units (7-4). The distance from the center to the y-axis is 3 units (since the radius extends 3 units to the left).

Formulating the Equation of the Circle

Now that we have determined the center and radius of the circle, we can formulate its equation. The standard form of a circle's equation is:

( (x - h)^2 (y - k)^2 r^2 )

Where (h, k) represents the center of the circle and r is the radius. Substituting the values we have found:

( (x - 3)^2 (y - 7)^2 3^2 )

which simplifies to:

( (x - 3)^2 (y - 7)^2 9 )

Key Insights and Further Exploration

This problem demonstrates the application of basic geometry and algebra. By understanding the properties of lines and circles, you can solve various geometric problems involving shapes and their interactions with lines. Here are some additional key insights:

The equation of a circle is dependent on its center and radius, as shown in the standard form. When dealing with problems that involve more than one geometrical shape (like lines and circles), it's crucial to identify the relevant distances and relationships. Practicing similar problems can significantly enhance your understanding and problem-solving skills.

Conclusion

By following the steps outlined in this guide, you can determine the equation of a circle that touches specific lines. This process combines fundamental concepts from algebra and geometry, making it a valuable skill for mathematicians and students alike. Whether you're a beginner or an advanced learner, mastering these techniques will help you tackle more complex problems and deepen your understanding of mathematical concepts.