Circle Equation Find Center And Radius For X^2+y^2-14x+12y+69=0

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In the realm of analytic geometry, circles hold a fundamental place. Their elegant symmetry and consistent properties make them a cornerstone of mathematical understanding. Among the many ways to represent a circle, the general equation form provides a powerful tool for analysis. In this article, we embark on a journey to dissect the equation x2+y2−14x+12y+69=0x^2 + y^2 - 14x + 12y + 69 = 0 and extract its defining characteristics: the center (h,k)(h, k) and the radius rr. We will delve into the process of transforming the general equation into the standard form, a transformation that illuminates the circle's essential features.

Transforming to Standard Form: Completing the Square

The given equation, x2+y2−14x+12y+69=0x^2 + y^2 - 14x + 12y + 69 = 0, is in the general form of a circle's equation. To readily identify the center and radius, we need to convert this equation into the standard form: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2. This transformation hinges on a technique known as completing the square. Completing the square allows us to rewrite quadratic expressions in a form that reveals the squared terms, which are crucial for identifying the circle's center and radius.

The Art of Completing the Square: A Step-by-Step Guide

  1. Group Like Terms: Begin by grouping the xx terms and the yy terms together, and move the constant term to the right side of the equation:

    (x2−14x)+(y2+12y)=−69(x^2 - 14x) + (y^2 + 12y) = -69

  2. Complete the Square for x: To complete the square for the xx terms, take half of the coefficient of the xx term (-14), square it ((-7)^2 = 49), and add it to both sides of the equation. This ensures that the equation remains balanced.

    (x2−14x+49)+(y2+12y)=−69+49(x^2 - 14x + 49) + (y^2 + 12y) = -69 + 49

  3. Complete the Square for y: Similarly, for the yy terms, take half of the coefficient of the yy term (12), square it ((6)^2 = 36), and add it to both sides of the equation.

    (x2−14x+49)+(y2+12y+36)=−69+49+36(x^2 - 14x + 49) + (y^2 + 12y + 36) = -69 + 49 + 36

  4. Factor and Simplify: Now, factor the quadratic expressions in xx and yy as perfect squares and simplify the right side of the equation.

    (x−7)2+(y+6)2=16(x - 7)^2 + (y + 6)^2 = 16

Decoding the Standard Form: Unveiling the Circle's Secrets

We have successfully transformed the general equation into the standard form: (x−7)2+(y+6)2=16(x - 7)^2 + (y + 6)^2 = 16. Comparing this to the general standard form, (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2, we can directly identify the circle's center and radius.

  • Center (h, k): By comparing the equations, we see that h=7h = 7 and k=−6k = -6. Therefore, the center of the circle is (7,−6)(7, -6).
  • Radius (r): The right side of the equation, 16, represents r2r^2. Taking the square root of 16, we find that the radius r=4r = 4.

The Center of the Circle: A Geometric Anchor

Having determined the center of the circle to be (7,−6)(7, -6), we have located a crucial point. The center serves as the geometric anchor of the circle, the point from which all points on the circle are equidistant. It is the heart of the circle's symmetry, the point around which the circle gracefully curves. Understanding the center's coordinates is paramount in visualizing the circle's position within the coordinate plane. It allows us to sketch the circle's outline and grasp its spatial relationship with other geometric figures. The center is more than just a point; it is the foundation upon which the circle's existence is built. Its coordinates, (7, -6), provide a precise location, a fixed reference point that defines the circle's place in the vast expanse of the coordinate system. This precise knowledge empowers us to analyze the circle's properties with greater accuracy and confidence.

The Radius of the Circle: A Measure of Extent

The radius, calculated as r=4r = 4, quantifies the circle's extent. It represents the constant distance from the center to any point on the circle's circumference. In essence, the radius is the measure of the circle's reach, its span in all directions from the central point. A larger radius signifies a larger circle, one that encompasses a greater area. Conversely, a smaller radius indicates a more compact circle, one that is closely gathered around its center. The radius is not merely a numerical value; it is a geometric descriptor, a tangible representation of the circle's size. It allows us to compare circles, to understand their relative proportions, and to appreciate the diversity of circular forms. A radius of 4 units provides a clear mental image of the circle's dimensions, its embrace of the space around its center. This understanding is fundamental to grasping the circle's overall characteristics and its role in geometric relationships.

The Significance of Standard Form: A Gateway to Understanding

The transformation of the general equation into the standard form, (x−7)2+(y+6)2=16(x - 7)^2 + (y + 6)^2 = 16, is not merely a mathematical manipulation; it is a gateway to understanding. The standard form is a revealing form, one that directly exposes the circle's key attributes: the center and the radius. It transforms an opaque expression into a transparent representation, allowing us to see the circle's inherent structure with clarity. The standard form is a geometric Rosetta Stone, a key to deciphering the circle's code. It provides a direct correspondence between the equation's components and the circle's physical properties. The values of hh and kk immediately tell us the center's coordinates, while the value of r2r^2 instantly unveils the radius. This directness is invaluable in problem-solving, in visualization, and in the general appreciation of circles as geometric objects. The standard form is a testament to the power of mathematical notation, its ability to encapsulate complex information in a concise and readily accessible form. Through the standard form, we gain a profound understanding of the circle's essence, its position, and its extent.

Visualizing the Circle: A Geometric Picture

With the center (7,−6)(7, -6) and the radius r=4r = 4 in hand, we can now construct a vivid geometric picture of the circle. Imagine a point located 7 units to the right of the origin and 6 units below it. This is the center, the circle's anchor. Now, visualize a circle drawn around this point, extending 4 units in all directions. This is the circle defined by the equation x2+y2−14x+12y+69=0x^2 + y^2 - 14x + 12y + 69 = 0. The visualization is crucial for solidifying our understanding. It transforms abstract equations into concrete forms, allowing us to see the circle as a tangible entity within the coordinate plane. We can imagine points tracing the circle's circumference, maintaining a constant distance of 4 units from the center. We can picture the circle's symmetry, its graceful curve around its central point. The geometric picture enhances our intuition, providing a visual foundation for our mathematical knowledge. It allows us to connect the equation to the shape, to see the circle as a whole, rather than a collection of symbols. This holistic understanding is the hallmark of true mathematical mastery.

Applications and Extensions: The Circle's Enduring Relevance

The study of circles extends far beyond the confines of academic exercises. Circles are fundamental shapes that appear throughout the natural world, from the orbits of planets to the ripples in a pond. Their properties are crucial in fields as diverse as engineering, physics, and computer graphics. Understanding the equation of a circle, and the techniques for analyzing it, is a skill that has enduring relevance. Circles are the basis for many technological innovations, from the wheels that propel our vehicles to the lenses that focus our vision. Their geometric properties are exploited in the design of bridges, the construction of buildings, and the creation of digital images. The ability to determine a circle's center and radius is not just a mathematical exercise; it is a practical skill with real-world applications. The circle's enduring presence in both the natural and the man-made world underscores its importance as a subject of study. Its simple elegance belies its profound utility, making it a cornerstone of mathematical understanding and a vital tool for scientific and technological progress.

Conclusion: The Circle's Equation Decoded

In conclusion, by employing the technique of completing the square, we successfully transformed the general equation x2+y2−14x+12y+69=0x^2 + y^2 - 14x + 12y + 69 = 0 into the standard form (x−7)2+(y+6)2=16(x - 7)^2 + (y + 6)^2 = 16. This transformation allowed us to identify the circle's center as (7,−6)(7, -6) and its radius as r=4r = 4. This exercise highlights the power of algebraic manipulation in revealing geometric properties. The standard form of a circle's equation provides a direct and insightful representation of its key characteristics, allowing us to visualize and analyze circles with greater clarity and confidence. The journey from the general equation to the identification of the center and radius is a testament to the beauty and utility of mathematics in unraveling the mysteries of the geometric world. By mastering these techniques, we equip ourselves with the tools to explore the rich and fascinating realm of circles and their applications.