Problem 17 - Algebra for College Students

Question

For Problems \(15-32\), find the center and the length of a radius of each of the circles. $$ (x+1)^{2}+(y+8)^{2}=12 \quad(-1,-8) ; r=2 \sqrt{3} $$

Step-by-Step Solution

Answer

Center: (-1, -8); Radius: \(2\sqrt{3}\)

Step 1: Identify the Standard Form of Circle Equation

The general form for the equation of a circle is \[(x - h)^2 + (y - k)^2 = r^2,\]where \((h, k)\) is the center of the circle and \(r\) is the radius.

Step 2: Compare with Given Equation

The given equation is \[(x+1)^2+(y+8)^2=12.\]Write it in the form of the standard equation of the circle: compare \((x - (-1))^2 + (y - (-8))^2 = 12\) with \((x - h)^2 + (y - k)^2 = r^2\). Thus, \(h = -1\) and \(k = -8\), making the center \((-1, -8)\).

Step 3: Find the Radius

Identify from the equation \((x+1)^2+(y+8)^2=12\)that \(r^2 = 12\), therefore the radius \(r\) is calculated as follows:\[r = \sqrt{12}.\]

Step 4: Simplify the Radius

Simplify \(\sqrt{12}\):\[r = \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}.\]Thus the radius \(r = 2\sqrt{3}\).

Key Concepts

Center of the Circle

When we talk about the center of a circle, we're referring to the point that is equidistant from all points on the circle's edge. To find the center of a circle using its equation, we use the standard form. The standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) represents the center.

• In the given equation, \((x+1)^2 + (y+8)^2 = 12\), you can see that it is written in the form \((x -(-1))^2 + (y -(-8))^2 = 12\).
• This comparison tells us that the center \((h, k)\) is located at \((-1, -8)\).

Understanding the center is crucial because it allows you to determine where the circle is positioned in the coordinate plane.

Radius Calculation

The radius of a circle is the distance from the center of the circle to any point on its edge. In mathematical terms, it is denoted as \(r\) in the circle's equation. This is depicted as \((x - h)^2 + (y - k)^2 = r^2\) in the standard form.

• From the equation \((x+1)^2 + (y+8)^2 = 12\), we identify \(r^2\) as 12.
• To find the radius \(r\), we take the square root of 12, resulting in \(r = \sqrt{12}\).
• By simplifying \(\sqrt{12}\), we break it down to \(\sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3}\).
• Since \(\sqrt{4} = 2\), the radius simplifies to \(2\sqrt{3}\).

This knowledge is vital as it gives us the circle's size and helps to visualize its scale on a graph.

Standard Form of a Circle Equation

The equation of a circle is most commonly presented in its standard form, which is \((x - h)^2 + (y - k)^2 = r^2\). This form makes it easy to identify both the center and the radius of the circle directly from the equation.

• The \((h, k)\) pair within the equation indicates the coordinates of the circle's center, making it straightforward to pinpoint the circle's location in the Cartesian plane.
• The \(r^2\) portion helps to determine the circle's radius by taking the square root, providing insight into the circle's extent.

Converting equations to this form simplifies the process of understanding and using circle equations in mathematical problems and real-world applications. This method allows for easier graphing and analysis of the circle's properties.