Finding The Equation Of A Circle Center And Point Given
In the realm of analytical geometry, circles hold a fundamental position. Defining a circle requires knowing its center and radius, which dictates its position and size on a coordinate plane. The equation of a circle provides a concise mathematical representation, allowing us to analyze and manipulate circles effectively. This article delves into how to determine the equation of a circle when given a point on the circle and its center, utilizing the distance formula and the standard form of a circle's equation. We will dissect a specific problem: finding the equation of a circle that contains the point (-2, 8) and has its center at (4, 0). This exploration will not only provide a step-by-step solution but also reinforce the underlying principles and formulas involved in circle geometry.
Understanding the Standard Equation of a Circle
The journey to finding the equation of a circle begins with understanding its standard form. The standard equation of a circle with center (h, k) and radius r is given by:
(x - h)² + (y - k)² = r²
This equation is derived directly from the Pythagorean theorem and the definition of a circle as the set of all points equidistant from a center. Here, (x, y) represents any point on the circle, (h, k) represents the coordinates of the center, and r denotes the radius. The radius is the constant distance between the center and any point on the circle. Recognizing this form is crucial because it allows us to easily extract the center and radius of a circle directly from its equation, and conversely, to construct the equation if we know the center and radius. Mastering this standard form is the cornerstone of solving circle-related problems in coordinate geometry. Understanding the interplay between the center, radius, and points on the circle provides a robust foundation for tackling more complex problems involving tangents, chords, and intersections of circles.
Applying the Distance Formula
The distance formula is a critical tool in coordinate geometry, particularly when dealing with circles. It allows us to calculate the distance between any two points in a coordinate plane, which is essential for determining the radius of a circle when we know the center and a point on the circumference. The distance formula is derived from the Pythagorean theorem and is expressed as:
√((x₂ - x₁)² + (y₂ - y₁)²)
Where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points. In the context of a circle, if we have the center (h, k) and a point (x, y) on the circle, the distance between these two points is the radius (r). Therefore, by applying the distance formula, we can find the radius using the coordinates of the center and the given point. This calculated radius is then used in the standard equation of a circle to fully define its equation. The distance formula serves as a bridge connecting the geometric concept of a radius with the algebraic representation of the circle's equation, making it an indispensable part of solving circle problems. The precision offered by the distance formula ensures that the calculated radius is accurate, leading to a correct equation of the circle. For complex geometric problems, the reliable calculation of distances is often the key to finding the correct solution.
Step-by-Step Solution
Let's apply our understanding to the problem at hand: finding the equation of a circle that contains the point (-2, 8) and has its center at (4, 0). This problem combines the concepts of the standard equation of a circle and the distance formula, providing a practical application of these mathematical tools.
1. Identify the Center and Point
The center of the circle is given as (4, 0), which corresponds to (h, k) in the standard equation. The point on the circle is (-2, 8), which will be our (x, y) for calculating the radius.
2. Use the Distance Formula to Find the Radius
We apply the distance formula to find the radius (r), which is the distance between the center (4, 0) and the point (-2, 8):
r = √((-2 - 4)² + (8 - 0)²) r = √((-6)² + (8)²) r = √(36 + 64) r = √100 r = 10
Thus, the radius of the circle is 10 units.
3. Plug the Values into the Standard Equation
Now that we have the center (h, k) = (4, 0) and the radius r = 10, we can plug these values into the standard equation of a circle:
(x - h)² + (y - k)² = r² (x - 4)² + (y - 0)² = 10² (x - 4)² + y² = 100
This final equation represents the circle that contains the point (-2, 8) and has a center at (4, 0). The step-by-step approach ensures that each component of the circle's equation is accurately determined, leading to the correct representation of the circle in the coordinate plane. The systematic application of the distance formula and the standard equation demonstrates a powerful method for solving circle-related problems in geometry.
Analyzing the Answer Choices
Now, let’s analyze the given answer choices to identify the correct equation:
A. (x - 4)² + y² = 100 B. (x - 4)² + y² = 10 C. x² + (y - 4)² = 10 D. x² + (y - 4)² = 100
By comparing our derived equation, (x - 4)² + y² = 100, with the answer choices, we can clearly see that option A matches our result. The other options either have an incorrect radius (10 instead of 100) or an incorrect center. Option B has the correct center but an incorrect radius squared value, while options C and D have the center incorrectly placed along the y-axis instead of the x-axis. This comparison highlights the importance of correctly applying the standard equation and the distance formula. Careful attention to the signs and values of the center coordinates and the radius is crucial for arriving at the correct equation. Analyzing answer choices in this manner not only confirms our solution but also reinforces our understanding of how the parameters in the equation relate to the circle's properties.
Conclusion
In conclusion, we successfully determined the equation of a circle that contains the point (-2, 8) and has a center at (4, 0) by employing the distance formula and the standard equation of a circle. The step-by-step process involved finding the radius using the distance formula and then substituting the center coordinates and the radius into the standard equation. This approach not only solved the specific problem but also illustrated a general method for finding the equation of a circle given its center and a point on the circumference. The correct equation, (x - 4)² + y² = 100, was identified by carefully comparing the derived equation with the provided answer choices. Mastering these fundamental concepts and techniques is essential for tackling more advanced problems in analytical geometry and related fields. Understanding the relationship between the geometric properties of a circle and its algebraic representation allows for a deeper comprehension of mathematical principles and their applications. Further practice with similar problems will solidify these skills and enhance problem-solving abilities in mathematics.
