Problem 61 - Intermediate Algebra

Question

Write each equation in standard form, if it is not already so, and graph it. If the graph is a circle, give the coordinates of its center and its radius. If the graph is a parabola, give the coordinates of its vertex. $$ (x-2)^{2}+y^{2}=25 $$

Step-by-Step Solution

Answer

The equation represents a circle with center at (2, 0) and radius 5.

Step 1: Recognize the Equation Type

The equation \((x-2)^2 + y^2 = 25\) is of the form \((x-h)^2 + (y-k)^2 = r^2\), which represents the equation of a circle.

Step 2: Identify the Standard Form Elements

In the equation \((x-2)^2 + y^2 = 25\), we compare it to the general circle equation \((x-h)^2 + (y-k)^2 = r^2\). Here:\(h = 2\)\(k = 0\)\(r^2 = 25\) which means \(r = \sqrt{25} = 5\).

Step 3: Determine the Center and Radius

The center of the circle, \((h, k)\), is \((2, 0)\) and the radius \(r\) is \(5\).

Step 4: Graph the Circle

To graph the circle, plot the center at \((2, 0)\). From this point, use the radius to draw a circle by ensuring a constant distance of 5 units from the center in all directions.

Key Concepts

Standard Form Equation

The standard form equation is a neat way to write equations of various geometric figures. For circles, the standard form is:

• \((x-h)^2 + (y-k)^2 = r^2\)

Here, \(h\) and \(k\) are the coordinates of the center of the circle, and \(r\) is the radius.
By writing the equation in this form, it's easy to identify key features of the circle. For any circular graph, the standard form instantly tells us where the center is and how big the circle is. This is crucial for graphing it accurately. The equation \( (x-2)^2 + y^2 = 25 \) is already in standard form, as it fits the necessary structure with distinct values for \(h\), \(k\), and \(r^2\). This makes pinpointing the circle’s center and radius straightforward.

Circle Equation

A circle equation is specifically tailored to describe all the points that make up a circle on a coordinate plane. Given in standard form, the equation

• \((x-h)^2 + (y-k)^2 = r^2\)

ensures that we understand a circle as the set of all points where the distance from the center, \(h, k\), is consistently equal to the radius, \(r\).
In our exercise, with the equation \((x-2)^2 + y^2 = 25\), the circle's constant measure from the center to any point on its circumference is its radius, determined by taking the square root of 25, producing a radius of 5 units.
The equation offers a quick and easy method to sketch out this round shape, guiding us seamlessly in plotting points and ensuring they all lie equidistantly from the center, forming a perfect circle.

Center and Radius of a Circle

Identifying the center and radius from a circle’s equation allows you to map out the circle easily on a graph.
The coordinates

• Center: \((h, k)\)
• Radius: \(r\)

plug directly into the circle's standard form equation. In \((x-2)^2 + y^2 = 25\), the center is found by equating:

• \((h, k) = (2, 0)\)

To find the radius, observe that the equation is set equal to \(r^2\), here \(r^2 = 25\), which means:

• \(r = 5\)

With this knowledge, you can easily draw the circle by starting at point \((2, 0)\) and measuring 5 units outward in every direction, marking each point along the distance to complete the circle. This method ensures accuracy and uniformity in your graphing.