Finding The Center Of A Circle From Its Equation A Step-by-Step Guide

In the realm of mathematics, particularly in coordinate geometry, circles hold a fundamental place. Understanding the equation of a circle is crucial for various applications, from basic geometry problems to more advanced concepts in calculus and analytical geometry. This article delves into the equation of a circle and how to extract key information, specifically the center, from its standard form. We will address the question: What is the center of a circle represented by the equation $(x+9)^2+(y-6)^2=10^2$?, guiding you through the process step-by-step and ensuring a clear understanding of the underlying principles.

The Standard Equation of a Circle

The standard equation of a circle in the Cartesian coordinate system is a powerful tool for representing circles and extracting information about their properties. This equation is given by:

$(x - h)^2 + (y - k)^2 = r^2$

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 the definition of a circle as the set of all points equidistant from a central point. The distance between any point $(x, y)$ on the circle and the center $(h, k)$ is always equal to the radius $r$. The standard equation provides a concise way to express this relationship algebraically.

Understanding the standard equation is the cornerstone of solving problems related to circles. It allows us to easily identify the center and radius of a circle if the equation is given, and conversely, to write the equation of a circle if the center and radius are known. This forms the basis for numerous geometric constructions, calculations, and analytical problem-solving techniques.

Identifying the Center from the Equation

The equation $(x - h)^2 + (y - k)^2 = r^2$ directly reveals the center of the circle as the point $(h, k)$. The key to correctly identifying the center is to pay close attention to the signs within the parentheses. If the equation has the form $(x + a)^2$, it implies that $h = -a$, and if it has the form $(y + b)^2$, it implies that $k = -b$. Conversely, if the equation has the form $(x - a)^2$, it implies that $h = a$, and if it has the form $(y - b)^2$, it implies that $k = b$. This sign convention is crucial for accurately determining the center's coordinates.

Consider the given equation: $(x+9)^2+(y-6)^2=10^2$. To find the center, we need to match this equation with the standard form $(x - h)^2 + (y - k)^2 = r^2$. We can rewrite $(x + 9)^2$ as $(x - (-9))^2$, which tells us that $h = -9$. Similarly, $(y - 6)^2$ directly gives us $k = 6$. Therefore, the center of the circle is $(-9, 6)$. This straightforward comparison to the standard form highlights the elegance and utility of the equation in quickly identifying the circle's center.

This method of extracting the center's coordinates by comparing the given equation to the standard form is a fundamental skill in coordinate geometry. It not only simplifies the process of finding the center but also enhances our understanding of how algebraic representations connect to geometric properties. By mastering this technique, you can confidently tackle a wide range of circle-related problems.

Applying the Concept to the Given Equation: $(x+9)^2+(y-6)^2=10^2$

Now, let's apply our understanding to the specific equation provided: $(x+9)^2+(y-6)^2=10^2$. Our goal is to determine the center of the circle represented by this equation. As we discussed earlier, the standard form of a circle's equation is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ represents the center.

Comparing the given equation to the standard form, we can see that the $x$ term is $(x + 9)^2$. To match the $(x - h)^2$ format, we can rewrite $(x + 9)^2$ as $(x - (-9))^2$. This tells us that $h = -9$. For the $y$ term, we have $(y - 6)^2$, which directly corresponds to the $(y - k)^2$ format. Therefore, $k = 6$.

Thus, the center of the circle represented by the equation $(x+9)^2+(y-6)^2=10^2$ is $(-9, 6)$. The radius of the circle is given by the square root of the constant term on the right side of the equation, which is $\sqrt{10^2} = 10$, but the question specifically asks for the center.

This step-by-step application of the standard equation demonstrates how easily we can extract the center's coordinates. By carefully comparing the given equation to the standard form and paying attention to the signs, we can accurately determine the center. This skill is essential for solving various problems involving circles and their properties.

Why the Other Options Are Incorrect

To further solidify our understanding, let's examine why the other options provided are incorrect. This will reinforce the importance of correctly interpreting the standard equation of a circle.

• Option B: $(-6, 9)$ If the center were $(-6, 9)$, the equation of the circle would be $(x + 6)^2 + (y - 9)^2 = 10^2$. This clearly does not match the given equation of $(x+9)^2+(y-6)^2=10^2$. The values within the parentheses are switched and have incorrect signs.

• Option C: $(6, -9)$ If the center were $(6, -9)$, the equation of the circle would be $(x - 6)^2 + (y + 9)^2 = 10^2$. Again, this does not match the given equation. The signs are incorrect, and the values are not consistent with the given equation.

• Option D: $(9, -6)$ If the center were $(9, -6)$, the equation of the circle would be $(x - 9)^2 + (y + 6)^2 = 10^2$. This option also does not match the given equation. The signs within the parentheses are reversed, leading to an incorrect center.

By systematically eliminating these incorrect options, we reinforce the correct method of extracting the center from the equation. This process of elimination highlights the precision required when working with the standard equation of a circle and emphasizes the significance of paying attention to the signs and order of terms.

Conclusion

In conclusion, the center of the circle represented by the equation $(x+9)^2+(y-6)^2=10^2$ is A. $(-9, 6)$. This determination was made by comparing the given equation to the standard form of a circle's equation, $(x - h)^2 + (y - k)^2 = r^2$, and correctly identifying the values of $h$ and $k$. We also explored why the other options were incorrect, reinforcing the importance of accurate interpretation and application of the standard equation.

Understanding the equation of a circle and its properties is fundamental to success in coordinate geometry and related mathematical fields. By mastering the techniques discussed in this article, you will be well-equipped to solve a variety of problems involving circles, their centers, radii, and equations. This knowledge forms a solid foundation for further exploration of geometric concepts and their applications in diverse areas of mathematics and beyond.