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Chapter 11: Problem 29

(a) find the center and radius, then (b) graph each circle. $$ (x+5)^{2}+(y+3)^{2}=1 $$

Short Answer

Expert verified
Center: (-5, -3), Radius: 1

Step by step solution

01

Identify the standard form of a circle equation

The given equation is \( (x+5)^{2}+(y+3)^{2}=1 \). This is in the standard form of a circle equation: \( (x-h)^{2} + (y-k)^{2} = r^{2} \) where \( (h, k) \) is the center and \( r \) is the radius.
02

Find the center of the circle

Compare the given equation with the standard form. Here, \( h=-5 \) and \( k=-3 \), so the center of the circle is \( (-5, -3) \).
03

Determine the radius of the circle

From the standard form, we see that \( r^{2} = 1 \). Taking the square root of both sides, \( r = \sqrt{1} = 1 \). Thus, the radius is \( 1 \).
04

Graph the circle

To graph the circle, plot the center \( (-5, -3) \). From this point, measure a distance of \( 1 \) unit in all directions (up, down, left, and right) to draw the circumference of the circle.

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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
To understand circles well, let's start with their standard equation form. The equation of a circle is given in the form \((x-h)^2 + (y-k)^2 = r^2\). Here, \((h, k)\) represents the center of the circle, and \(r\) is the radius. This form helps us quickly identify and work with the key properties of a circle, making calculations easier.

In simple terms:
  • \( (h, k) \) is the center.
  • \( r \) is the radius.
This standard form framework aids greatly in solving problems involving circles.
Finding the Center
Once we have the circle's equation in the standard form, \((x-h)^2 + (y-k)^2 = r^2\), locating the center is straightforward.

For example, given the equation \((x+5)^2 + (y+3)^2 = 1\), we need to match it against the general form. By comparing terms:
  • \(h = -5\) (since \(x+5\) is \(x - (-5)\))
  • \(k = -3\) (since \(y+3\) is \(y - (-3)\))
Hence, the center of this circle is \((-5, -3)\). Identifying these shifts both horizontally and vertically gives us the center's coordinates.
Determining the Radius
After determining the center, the next steps involve finding the radius, denoted by \(r\). The equation \((x-h)^2 + (y-k)^2 = r^2\) shows that \(r^2\) is the value on the right side of the equation.

For our given example, \((x+5)^2 + (y+3)^2 = 1\), we see that \(r^2 = 1\). To find the radius \(r\), we take the square root of both sides:
  • \(r^2 = 1\)
  • \(r = \sqrt{1} = 1\)
So, the radius for this circle is \(1\).
Graphing Circles
Graphing circles involves plotting their center first and then using the radius to draw the circumference.

For our exercise, the steps are:
  • Plot the center \((-5, -3)\).
  • From this center, measure out the radius distance in all directions (up, down, left, right). For instance, a 1-unit distance in each direction from \((-5, -3)\).
  • Sketch a smooth curve connecting these points to form the circle.
While doing this, ensure accuracy in measurement to depict the right size and position of the circle. Remember, staying precise helps in representing the circle accurately.

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Most popular questions from this chapter

Solve the system of equations by using substitution. $$ \left\\{\begin{array}{l} y=x^{2}-4 \\ y=x-4 \end{array}\right. $$

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