Question

Find the center and radius of each circle. Graph. $$(x+5)^{2}+(y-2)^{2}=49$$

Short answer

The center of the circle is \((-5, 2)\) and the radius is \(7\). The circle can be graphed with these parameters.

Step-by-step solution

Identify the Standard Form of a Circle

Recognize that the equation \((x+5)^{2}+(y-2)^{2}=49\) is already in the standard form of a circle's equation, which is \((x-h)^2 + (y-k)^2 = r^2\) where \((h, k)\) is the center of the circle and \(r\) is the radius.

Determine the Center of the Circle

Comparing the equation \((x+5)^{2}+(y-2)^{2}=49\) to the standard form, identify the center \((h, k)\) of the circle by noting the changes in sign inside the parentheses. The center is \((-5, +2)\).

Calculate the Radius of the Circle

Observe that the number on the right-hand side of the equation, \(49\), is the square of the radius \(r^2\). To find the radius, take the square root of \(49\), which is \(r = 7\).

Graph the Circle

Plot the center of the circle at \((-5, 2)\) on the coordinate plane. Then, using a compass, draw a circle with the radius of \(7\) units around the center. Make sure the circle passes through points that are \(7\) units away from the center in all directions.

Key concepts

Circle Equations

Understanding the equation of a circle is crucial when it comes to graphing and analyzing the properties of these round shapes. The standard form of a circle's equation is expressible as \((x-h)^2 + (y-k)^2 = r^2\), where the point \((h, k)\) is the center of the circle, and \(r\) stands for the radius.

When given an equation like \((x+5)^2 + (y-2)^2 = 49\), we can discern the circle's features by carefully comparing the equation to this standard form. The center is found by taking the opposite signs of the numbers inside the parentheses, yielding \((-5, 2)\) in this case. The \(r^2\) term, here represented by \(49\), tells us the radius of the circle after taking the square root, which in this example, is \(7\).

The beauty of the standard form is that it immediately reveals these key characteristics of the circle. While the equation \((x+5)^2 + (y-2)^2 = 49\) might look complex at first glance, recognizing its structure allows us to easily deduce both the center and the radius.

Graphing Circles

Putting a circle on a graph does not have to be a daunting task once you break it down into a few simple steps. Starting with the standard form of a circle's equation, such as \((x+5)^2 + (y-2)^2 = 49\), we first pinpoint the center at \((-5, 2)\).

The graphing process involves plotting this central point on the coordinate grid first. From there, you will measure outwards in every direction the distance equal to the radius, which we've determined to be \(7\) units for our example. Make sure to mark several points exactly \(7\) units from the center, to ensure an even and accurate circle.

Drawing the Circle

Once you have the center and peripheral points, you can use a compass or carefully sketch freehand to connect these points and form the circle. For precision, especially in mathematical contexts, using a compass is recommended to maintain consistent curvature.

Radius and Center of a Circle

In the realm of geometry, knowing the center and radius of a circle provides immense insight into its size and position. The center is the fixed point from which every point on the circumference is equidistant. Meanwhile, the radius is the distance from the center to any point on the edge of the circle.

In our textbook example, we extracted the circle's center \((-5, 2)\) and radius \(7\) by analyzing the standard form equation. These two attributes can inform a plethora of other aspects such as the circle's area, \(\pi r^2\), and circumference, \(2\pi r\).

Real-World Applications

The concepts of radius and center are not merely academic; they are applied in various real-world scenarios, from designing wheels to predicting the orbits of celestial bodies. A deep understanding of these fundamental characteristics of circles can thus provide foundational knowledge for various scientific and engineering disciplines.