Equation Of A Circle Center (-2 10) And Point (10 5) Explained

by ADMIN 63 views

In the realm of mathematics, particularly in coordinate geometry, understanding the equation of a circle is fundamental. This article delves into a specific problem concerning the equation of a circle, providing a detailed explanation and solution. We will dissect the problem step by step, ensuring clarity and comprehension for readers of all levels. Our main focus will be on identifying the correct equation for circle C, which has its center at (-2, 10) and passes through the point P(10, 5). This exploration will not only solve the problem at hand but also solidify your understanding of the general equation of a circle and its components.

Decoding the Circle Equation

The equation of a circle in the Cartesian plane is a powerful tool that allows us to describe and analyze circles using algebraic expressions. The general form of the equation is given by:

(x - h)² + (y - k)² = r²

Where:

  • (x, y) represents any point on the circumference of the circle.
  • (h, k) represents the coordinates of the center of the circle.
  • r represents the radius of the circle.

This equation is derived from the Pythagorean theorem and embodies the definition of a circle: the set of all points equidistant (the radius) from a central point. Understanding each component of this equation is crucial for solving problems related to circles. The values of 'h' and 'k' directly determine the circle's position on the coordinate plane, while 'r' dictates its size. The squared terms reflect the application of the Pythagorean theorem in calculating distances. By manipulating this equation, we can determine various properties of a circle, such as its center, radius, and points lying on its circumference. This foundational knowledge is indispensable for tackling more complex problems in geometry and calculus.

Applying the Formula to Our Problem

In our specific problem, we are given the center of circle C as (-2, 10), which means h = -2 and k = 10. We are also given a point P(10, 5) that lies on the circle. To find the equation of the circle, we need to determine the radius, r. The radius is the distance between the center of the circle and any point on its circumference. We can calculate this distance using the distance formula:

r = √[(x₂ - x₁)² + (y₂ - y₁)²]

Where (x₁, y₁) is the center of the circle and (x₂, y₂) is the point P. Plugging in the values, we get:

r = √[(10 - (-2))² + (5 - 10)²] r = √[(12)² + (-5)²] r = √(144 + 25) r = √169 r = 13

Therefore, the radius of circle C is 13. Now that we have the center (h = -2, k = 10) and the radius (r = 13), we can plug these values into the general equation of a circle:

(x - (-2))² + (y - 10)² = 13² (x + 2)² + (y - 10)² = 169

This final equation represents circle C. This step-by-step calculation highlights the importance of understanding the distance formula and its application in finding the radius. By substituting the calculated radius and the given center coordinates into the general equation, we arrive at the specific equation that defines our circle. This process underscores the interconnectedness of geometric concepts and algebraic representations.

Analyzing the Answer Choices

Now that we have derived the equation of circle C, let's analyze the given answer choices to identify the correct one:

A. (x - 2)² + (y + 10)² = 13 B. (x - 2)² + (y + 10)² = 169 C. (x + 2)² + (y - 10)² = 13 D. (x + 2)² + (y - 10)² = 169

Comparing our derived equation, (x + 2)² + (y - 10)² = 169, with the answer choices, we can clearly see that option D matches perfectly. Options A and C have the correct form for the center but incorrect values for the radius squared. Option B has the incorrect form for the center and the radius squared. Therefore, option D is the correct answer. This comparative analysis emphasizes the importance of carefully examining each component of the equation – the signs, the center coordinates, and the radius squared – to ensure accuracy. By systematically comparing our derived equation with the given options, we can confidently identify the correct answer and avoid common errors.

Common Mistakes to Avoid

When dealing with circle equations, several common mistakes can lead to incorrect answers. One frequent error is confusing the signs of the center coordinates. Remember that the equation uses (x - h) and (y - k), so if the center is at (-2, 10), the equation will have (x + 2) and (y - 10). Another common mistake is forgetting to square the radius. The equation uses r², so if the radius is 13, the equation should have 13², which is 169. Additionally, errors in calculating the distance between two points can lead to an incorrect radius. It's crucial to apply the distance formula accurately. To avoid these mistakes, it's helpful to write down the general equation of a circle and carefully substitute the given values. Double-checking your calculations, especially when dealing with signs and squares, can prevent errors. Understanding the underlying concepts and practicing with various examples can also help solidify your understanding and reduce the likelihood of making these common mistakes. By being mindful of these pitfalls, you can approach circle equation problems with greater confidence and accuracy.

Conclusion Mastering Circle Equations

In conclusion, understanding the equation of a circle is essential in coordinate geometry. By correctly applying the formula (x - h)² + (y - k)² = r², we can determine the equation of a circle given its center and a point on its circumference. In this specific problem, we found that the equation representing circle C, with a center at (-2, 10) and passing through the point P(10, 5), is (x + 2)² + (y - 10)² = 169. By carefully analyzing the problem, applying the relevant formulas, and avoiding common mistakes, we can confidently solve such problems. Mastering circle equations not only enhances your mathematical skills but also provides a solid foundation for tackling more advanced topics in geometry and calculus. This journey through the equation of a circle highlights the importance of precision, attention to detail, and a thorough understanding of fundamental concepts in mathematics.