Problem 23 - Algebra 2

Question

Find the center and radius of the circle with the given equation. Then graph the circle. $$ (x+3)^{2}+(y+7)^{2}=81 $$

Step-by-Step Solution

Answer

The center is \((-3, -7)\) and the radius is 9. Graph by plotting the center and using the radius.

Step 1: Recognize the standard form of a circle equation

The equation \((x+3)^2 + (y+7)^2 = 81\) is already in the standard form for a circle, which is \((x-h)^2 + (y-k)^2 = r^2\). Here, \((h, k)\) represents the center of the circle, and \(r\) is the radius.

Step 2: Identify the center of the circle

Compare the given equation \((x+3)^2 + (y+7)^2 = 81\) with the standard form \((x-h)^2 + (y-k)^2 = r^2\). To find \(h\) and \(k\), note that \(h\) is the number added to or subtracted from \(x\), and \(k\) is the number added to or subtracted from \(y\). This gives \((h, k) = (-3, -7)\). Thus, the center of the circle is \((-3, -7)\).

Step 3: Identify the radius of the circle

The given equation \((x+3)^2 + (y+7)^2 = 81\) can be compared to \((x-h)^2 + (y-k)^2 = r^2\), where \(r^2 = 81\). Taking the square root, we find \(r = \sqrt{81} = 9\). So, the radius of the circle is 9 units.

Step 4: Graph the circle

To graph the circle, start by plotting the center point \((-3, -7)\) on a coordinate plane. Next, use the radius of 9 units to plot several points around the center that are 9 units away in all directions (up, down, left, right, and diagonals). These points form the circumference of the circle. Connect these points smoothly to draw the circle.

Key Concepts

Standard Form of a Circle

The standard form of a circle's equation is a concise way to capture the essential attributes of a circle on a coordinate plane. This form is given by the equation:

• \((x-h)^2 + (y-k)^2 = r^2\)

Here,

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

When a circle's equation is arranged in this standard form, it straightforwardly reveals the center and the radius.
This helps in easily graphing the circle and understanding its position and size on the coordinate plane. Consider recognizing these components in any given circle equation by consistently comparing it to the standard form.

Center of a Circle

Understanding the center of a circle is fundamental to graphing and analyzing its properties on a coordinate plane.
The center is the point equidistant from all points on the circle's circumference.
You will find the coordinates of the center, \((h, k)\), when the circle's equation is in its standard form:

• \((x-h)^2 + (y-k)^2 = r^2\).

To determine the circle's center, simply identify what's being subtracted from \(x\) and \(y\) in the equation.

• For example, in the equation \((x+3)^2 + (y+7)^2 = 81\),
• the numbers +3 and +7 actually indicate the center at \((-3, -7)\).

These signs change because the standard form uses \((x-h)\) and \((y-k)\), and thus negate the respective signs involved in the equation.

Radius of a Circle

The radius of a circle is the distance from its center to any point on the perimeter of the circle.
In the standard circle equation form \((x-h)^2 + (y-k)^2 = r^2\), the radius is represented by \(r\).

• The term \(r^2\) is crucial because it appears directly in the equation.

To find the radius, you simply take the square root of the value found on the right side of the equation.
In our example, the equation \((x+3)^2 + (y+7)^2 = 81\) reveals \(r^2 = 81\).
Thus, the radius \(r\) is \(\sqrt{81} = 9\) units.

• Having the radius helps you sketch the circle accurately by indicating how far the circle's edge extends from its center.

Understanding the radius is also essential when calculating other properties of the circle, such as its circumference and area.