Problem 71 - 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. $$ (x-2)^{2}+(y-4)^{2}=36 $$

Step-by-Step Solution

Answer

The graph is a circle with center (2, 4) and radius 6.

Step 1: Identify the equation type

The given equation \((x-2)^2 + (y-4)^2 = 36\) is in the form \((x-h)^2 + (y-k)^2 = r^2\), which indicates that it represents a circle.

Step 2: Determine the center coordinates and radius

For a circle, the standard form is \((x-h)^2 + (y-k)^2 = r^2\). Comparing with the given equation \((x-2)^2 + (y-4)^2 = 36\), we identify \(h = 2\), \(k = 4\), and \(r^2 = 36\). Solving for \(r\), we find \(r = 6\). This means the center of the circle is at \((2, 4)\) and the radius is 6.

Step 3: Graph the circle

To graph the circle, plot the center at the point \((2, 4)\). From this center, draw a circle with a radius of 6. This means moving 6 units up, down, left, and right from the center to mark the circle's boundary. Connect these points smoothly to form the circle.

Key Concepts

Standard Form

In geometry, the equation of a circle is often expressed in what is known as the **standard form**.This form provides a clear and concise way to understand different attributes of the circle, such as its center and radius.The standard form of a circle equation is \[(x-h)^2 + (y-k)^2 = r^2\]where:

• \((h, k)\) are the coordinates of the circle's center.
• \(r\) represents the radius of the circle.

A circle equation in this form makes it easy to quickly identify these properties without additional calculation.For example, when looking at the equation \[ (x-2)^2 + (y-4)^2 = 36 \], we can immediately see it is in standard form, allowing us to move directly to deriving other circle properties like the center and radius.

Center Coordinates

Once a circle's equation is in standard form, you can easily find its center coordinates.The center of a circle represented by \[(x-h)^2 + (y-k)^2 = r^2\] is simply the ordered pair \((h, k)\).So for the equation \[(x-2)^2 + (y-4)^2 = 36\],we can see that:

• \(h = 2\), meaning the \(x\)-coordinate of the center is 2.
• \(k = 4\), meaning the \(y\)-coordinate of the center is 4.

Thus, the center of the circle is at the point \((2, 4)\).Knowing the center of a circle is crucial in both graphing and analyzing the circle's placement on the coordinate plane.

Radius

The radius of a circle is another key characteristic that tells us about the size of the circle.Once we have a circle's equation in standard form, finding the radius becomes straightforward.In the standard form equation:\[(x-h)^2 + (y-k)^2 = r^2\],\(r^2\) represents the square of the radius.To find the actual radius \(r\),you simply take the square root of \(r^2\).For the equation \[(x-2)^2 + (y-4)^2 = 36\], we see that \(r^2 = 36\).By taking the square root:\[r = \sqrt{36} = 6\]So, the radius of the circle is 6 units.This means that from the center of the circle, any point on the circle is 6 units away.The radius is also vital in drawing the circle as it helps define its size on a graph.