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Chapter 0: Problem 55

Determine the center and radius of each circle and sketch the graph. $$ (x+6)^{2}+y^{2}=36 $$

Short Answer

Expert verified
The center is \( (-6, 0) \) and the radius is 6.

Step by step solution

01

Identify the standard form of a circle

The equation of a circle in standard form is \( (x-h)^{2} + (y-k)^{2} = r^{2} \) where \( (h, k) \) is the center and \( r \) is the radius.
02

Rewrite the given equation in standard form

Compare \( (x+6)^{2} + y^{2} = 36 \) with the standard form equation. Here, \( h = -6 \), \( k = 0 \), and \( r^{2} = 36 \).
03

Identify the center

From the comparison, the center of the circle \( (h, k) \) is found to be \( (-6, 0) \).
04

Calculate the radius

Since \( r^{2} = 36 \, r = \sqrt{36}\), yielding \( r = 6 \).
05

Sketch the graph

Plot the center of the circle at \( (-6, 0) \). Draw a circle with radius 6 units around this point.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

standard form of a circle
The standard form of a circle's equation makes it simple to identify the circle's center and radius. It is written as \((x-h)^{2} + (y-k)^{2} = r^{2} \).
This format breaks down like this:
  • \(h\) and \(k\) provide the coordinates for the circle's center. \((h, k)\)
  • \(r\) represents the circle's radius
Standardizing helps because you can directly extract key information about the circle, simply by comparing it to this form.
center of a circle
Identifying the center of a circle is straightforward once you have the standard form of the equation. Compare your circle's equation to \((x-h)^{2} + (y-k)^{2} = r^{2} \). In the equation \((x+6)^{2} + y^{2} = 36 \),
  • The term \((x+6)^{2}\) shows that \(-6\) is the \(x\)-coordinate of the center (since \h\ is \(-6\) here)
  • The \(y^{2}\) term shows that \(k\) is \(0\)
So, the center is at \(-6, 0\).
This understanding allows easy plotting of the circle's center on a graph.
radius calculation
Finding the radius is just as easy as finding the center. The standard form equation \((x-h)^{2} + (y-k)^{2} = r^{2}\) aids in this:
From the equation \((x+6)^{2} + y^{2} = 36 \),
  • The right side (where \(36\) is) equals \(r^{2}\).
So, to find \(r\), take the square root of \(36\):
\[ r^{2} = 36 \]
Apply the square root to find \(\text{r = } \text{\textpm}6 \). Since radius is a length, we take \(r=6\).
This gives the distance from the center to any point on the circle.
graphing a circle
Graphing a circle involves a couple of simple steps once you have identified the center and radius.
  • First, plot the center \((-6, 0)\) on your graph.
  • From that center, measure a radius of 6 units in all directions.
For example, you'll mark points at:
  • 6 units to the right (\rightarrow (0, 0))
  • 6 units to the left (\rightarrow (-12, 0))
  • 6 units up (\rightarrow (-6, 6))
  • 6 units down (\rightarrow (-6, -6))
Once you have these reference points, draw a smooth curve connecting them to form your circle.
This visual representation makes understanding the circle's shape and size much easier.

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