Problem 39 - College Algebra

Question

Write the standard form of the equation of the circle with the given center and radius. $$ \text { Center }(-4,0), r=10 $$

Step-by-Step Solution

Answer

The standard form of the equation of the circle with the center at (-4,0) and radius 10 is \((x + 4)^2 + y^2 = 100\).

Step 1: Identify the given parameters

From the problem, we can ascertain that the center of the circle is \((-4, 0)\) and the radius of the circle is \(10\) units. This means that \(h = -4\), \(k = 0\), and \(r = 10\).

Step 2: Substitute values into the standard equation

Now we need to substitute the values of \(h = -4\), \(k = 0\), and \(r = 10\) into the equation. It then becomes \((x - (-4))^2 + (y - 0)^2 = 10^2\).

Step 3: Simplify the equation

Simplify the equation to its standard form. The equation then simplifies to \((x + 4)^2 + y^2 = 100\).

Key Concepts

Standard Form of the Circle Equation

The standard form of the equation of a circle is a crucial concept in geometry. It is a way to express a circle algebraically using its center and radius. The general form of a circle's equation in the standard form is: \\[ (x - h)^2 + (y - k)^2 = r^2 \]
where:

• \(h\) and \(k\) are the coordinates of the center of the circle, \((h, k)\).


• \(r\) is the radius of the circle.

Using this form, one can easily identify both the center and the radius of the circle just by looking at the equation. In this format, both the \((x - h)^2\) and \((y - k)^2\) components represent squared terms that measure the distance from any point \((x, y)\) on the circle to the center \((h, k)\) in the x and y directions, respectively. Therefore, this form helps ascertain the basic properties of the circle in a visually intuitive manner.

Center of the Circle

The center of a circle is a point that is equidistant from all points on the circle. In the standard form of the equation of a circle, the center is represented by the coordinates \((h, k)\). For our problem, the center provided is \((-4, 0)\), which means:

• \(h = -4\)


• \(k = 0\)

This implies that the circle is located on the coordinate plane such that its center is 4 units to the left of the origin along the x-axis, and 0 units vertically from the origin along the y-axis. Understanding the location of the center assists in graphing the circle accurately and helps with further geometric analysis, such as understanding the circle's symmetry and relation to other geometric figures.

Radius of the Circle

The radius of a circle is defined as the constant distance from the center of the circle to any point on its circumference. In the equation \\[ (x - h)^2 + (y - k)^2 = r^2 \]
\(r\) represents the radius.In this specific problem, the radius is given as \(r = 10\). This means every point on the circle is exactly 10 units away from the center \((-4, 0)\). The radius is critical not only for the equation but also for understanding the size and scale of the circle. By squaring the radius, as seen in the equation, it helps plot and confirm the span or reach of the circle on a coordinate plane. Knowing the radius allows us to establish boundaries and dimensions necessary for applications such as circle-sector calculations or solving real-world problems like mapping and design.