Problem 12 - Technical Mathematics with Calculus

Question

Find the center and radius of each circle. Graph. $$(x-2.22)^{2}+(y+7.16)^{2}=5.93$$

Step-by-Step Solution

Answer

The center of the circle is \( (2.22, -7.16) \) and the radius is \( \sqrt{5.93} \). The circle can be graphed by plotting the center and drawing a circle with the calculated radius.

Step 1: Identify the Center

The equation of the circle is in the standard form \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center of the circle and \( r \) is the radius. By comparing this to the given equation \( (x - 2.22)^2 + (y + 7.16)^2 = 5.93 \), we can identify the center as \( (h, k) = (2.22, -7.16) \) by observing the values being subtracted from \( x \) and \( y \) in the equation.

Step 2: Determine the Radius

The radius \( r \) of the circle is the square root of the constant on the right side of the equation. From the given equation \( (x-2.22)^{2} + (y+7.16)^{2} = 5.93 \), we can determine the radius by taking the square root of 5.93, which gives \( r = \sqrt{5.93} \).

Step 3: Graph the Circle

To graph the circle, first plot the center at \( (2.22, -7.16) \). Then, using the radius \( r = \sqrt{5.93} \), measure this distance out from the center in all directions to form the circle. Ensure that the circle is round and that all points on the circumference are equidistant from the center.

Key Concepts

Circle Equation

Understanding the equation of a circle is crucial for graphing it. The general form of a circle's equation is \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) denotes the circle's center and \( r \) represents its radius. In this equation, the quantities \( h \) and \( k \) are subtracted from \( x \) and \( y \) respectively, and then squared. The result of these squared differences must always equal the squared radius of the circle for any point \( (x, y) \) on the circumference.

To visualize, each point on the circle maintains the same distance, \( r \) from the center. This unique characteristic allows us to generate the perfect round shape of the circle by ensuring every point forms a circle with radius \( r \) around the center \( (h, k) \) on a coordinate plane.

Finding Circle Center and Radius

To find the center and radius of a circle from its equation, take a look at the standard form and identify \( h \) and \( k \) which correspond to the coordinates of the center. For the exercise \( (x-2.22)^{2} + (y+7.16)^{2} = 5.93 \), comparing it with the standard form, \( h \) is 2.22 and \( k \) is -7.16 (watch out for signs!), thus the center is \( (2.22, -7.16) \). The radius is the square root of the constant term on the equation's right side. In our example, this is \( r = \sqrt{5.93} \). Here's a practical tip: Always ensure to consider the square root of the entire constant term for the radius, as omitting or incorrectly squaring this value will lead to errors in further calculations and graphing.

Plotting a Circle on a Coordinate Plane

When plotting a circle on a coordinate plane, start by marking the center point. For the given circle with center \( (2.22, -7.16) \), place a dot precisely at that point. Next, use the radius \( r = \sqrt{5.93} \) as a measuring guide. From the center, this distance should be mapped in all directions—up, down, left, right, and diagonally. Think of it as drawing a 'boundary' that's always the same distance from the center.

One effective approach is to draw a horizontal and a vertical line passing through the center to create a cross. Then measure the radius on each of these lines to find four points that will lie on the circle. After plotting these guiding points, sketch the circle carefully to make sure it's round and that the center maintains equal distance to any point on the circumference. If done correctly, the circle should curve smoothly between these points, maintaining its symmetry and roundness.