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The Mysteries of Circle Centers: Decoding (x+9)2+(y−6)2=102
Circles are fascinating geometric shapes that have captured the imagination of artists, mathematicians, and philosophers for thousands of years. In addition to their aesthetic appeal, circles have many practical applications in fields such as engineering, physics, and architecture. However, circles also pose some challenging questions, especially when it comes to finding their centers. In this article, we will explore the mysteries of circle centers by decoding the equation (x+9)2+(y−6)2=102, which represents a specific circle in the Cartesian plane. By the end of this journey, you will have gained a deeper understanding of circles, coordinates, and calculations, and will be able to solve similar problems on your own.
Before we start unraveling the secrets of the circle (x+9)2+(y−6)2=102, let’s review some basic concepts of circles and coordinates. A circle can be defined as the set of points that are equidistant from a center point. The distance from the center to any point on the circle is called the radius, which is usually denoted by the letter r. Thus, if we know the center and the radius of a circle, we can easily plot the circle and measure its properties, such as area, circumference, and diameter. Coordinates, on the other hand, are pairs of numbers that represent the position of a point in the Cartesian plane, which is like a map with two perpendicular axes (x and y) that intersect at the origin (0,0). Each coordinate pair corresponds to a unique point in the plane, and vice versa. By using coordinates, we can describe the location and movement of objects in space, as well as calculate distances, angles, and slopes.
Now, let’s look at the equation (x+9)2+(y−6)2=102, which represents a circle with center (-9,6) and radius 10. How do we know that? How can we graph the circle and verify its properties? There are several steps involved in solving this problem, but if we follow them carefully, we should be able to succeed.
Step 1: Recognize the standard form of a circle equation. The general form of a circle equation is (x−h)2+(y−k)2=r2, where (h,k) is the center and r is the radius. However, in some cases, the equation may be presented in a different format, such as (x+9)2+(y−6)2=102. To convert this equation into the standard form, we need to isolate the terms that involve x and y on one side and the constant term on the other side. First, we expand the square of the binomials using the FOIL method (First, Outer, Inner, Last): (x+9)2=x2+18x+81 and (y−6)2=y2−12y+36. Then, we simplify the left-hand side by adding these two expressions and equating them to 102: x2+18x+y2−12y+117=0. Finally, we move the constant term to the right-hand side and group the x-terms and the y-terms together: x2+18x+y2−12y=-117. Now, we can identify the center and the radius by comparing the coefficients with the standard formula. Since the x-term contains 2h (which is 18), we have h=-9. Similarly, since the y-term contains 2k (which is -12), we have k=6. Finally, since the constant term is r2 (which is 100), we have r=10. Therefore, the circle (x+9)2+(y−6)2=102 has center (-9,6) and radius 10.
Step 2: Plot the center and label the radius. To graph the circle in the Cartesian plane, we first need to locate its center point (-9,6) on the x-axis and the y-axis. We can draw a cross at that point or simply mark the coordinates with their signs. Then, we need to measure the distance from the center to any point on the circle, which is the radius of the circle. We can use a ruler or a protractor to draw a line segment of length 10 units (or equivalent) from the center to the edge of the circle, in any direction. We can label this line as r=10 or simply write 10 next to it. Now, we have a reference frame for the circle and can proceed to graph it by connecting the points that satisfy the equation (x+9)2+(y−6)2=102.
Step 3: Use symmetry and algebra to find other points. One of the properties of circles is that they are symmetric with respect to their center. That means, any point that is equidistant from the center as another point on the circle must also lie on the circle. Moreover, since the circle has rotational symmetry, it repeats itself every 360 degrees (or 2π radians) around its center. This means that we only need to find a few points on the circle to draw it completely. But how can we find these points? One way is to use algebraic manipulation of the equation. Suppose we want to find a point (x,y) on the circle that has x-coordinate 1. That means, x+9=1, or x=-8. If we substitute this value into the equation, we get (−8+9)2+(y−6)2=102, which simplifies to 1+y2−12y=1. Solving for y, we get y=6±√19. Therefore, we have two points on the circle with x-coordinate -8 and y-coordinates 6+√19 and 6-√19. We can plot these points on both sides of the center (-9,6) and connect them to form a segment that intersects the circle at two other points. By repeating this process for other x-coordinates, such as -7, -6, 0 or 10, we can obtain more points on the circle and verify its shape and size.
Step 4: Check your work and interpret the results. Once we have graphed the circle and found several points on it, we can check our work by using geometric properties such as symmetry, Pythagorean theorem, or trigonometry. For example, we can confirm that the distance between any two points on the circle is equal to the radius, or that the diameter (or twice the radius) passes through the center. We can also calculate the area and circumference of the circle using well-known formulas, such as A=πr2 and C=2πr. Finally, we can interpret the results by thinking about the context of the problem, such as how the circle relates to a real-world scenario or a mathematical concept. For instance, we may ask ourselves questions like: What is the significance of the center and radius of the circle in the context of the equation? What other circles share similar properties or symmetries with this circle? How can we generalize or extend this problem to other cases?
By following these steps, we have decoded the mysteries of circle centers and revealed the hidden structure of the equation (x+9)2+(y−6)2=102. We have learned how to identify the center and radius of a circle, plot its points, and derive new points using symmetry and algebraic manipulation. We have also seen how circles relate to coordinates, distance, symmetry, and geometry in general. This knowledge can be useful in many areas of mathematics, science, and engineering, as well as in art and design. Moreover, this knowledge can deepen our appreciation and understanding of the beauty and complexity of circles and their role in shaping our world.