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Chapter 11: Problem 80

Find the center and the radius of the circle with the given equation. [ 10.2] $$x^{2}+y^{2}-6 x+4 y=3$$

Short Answer

Expert verified
Center: (3, -2), Radius: 4

Step by step solution

01

Rearrange the equation

Start by rearranging the given equation into a form that groups the x and y terms together: \[ x^2 - 6x + y^2 + 4y = 3 \]
02

Complete the square for the x terms

To complete the square for the x terms, take the coefficient of x (-6), divide it by 2, and then square it: \[ (-6/2)^2 = 9 \]Add and subtract this value inside the equation: \[ x^2 - 6x + 9 - 9 + y^2 + 4y = 3 \]
03

Complete the square for the y terms

To complete the square for the y terms, take the coefficient of y (4), divide it by 2, and then square it: \[ (4/2)^2 = 4 \]Add and subtract this value inside the equation: \[ x^2 - 6x + 9 + y^2 + 4y + 4 = 3 + 9 + 4 \]
04

Rewrite the equation

Combine the completed squares and simplify: \[ (x-3)^2 + (y+2)^2 = 16 \]This is the standard form of the equation of a circle.
05

Identify the center and the radius

From the standard form \[ (x-h)^2 + (y-k)^2 = r^2 \], where \( h = 3 \) and \( k = -2 \), and \( r^2 = 16 \), the center is (3, -2) and the radius is 4.

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Key Concepts

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

Completing the Square
Completing the square is a method used to convert a quadratic equation into a perfect square trinomial. This is especially useful for finding the standard form of a circle's equation.
To complete the square for a variable, follow these steps:
  • Take the coefficient of the linear term (the x term or y term), divide it by 2, and then square it.
  • Add and subtract this squared value in the equation. This helps in forming a perfect square trinomial.
Let's see this in action with the equation: \[ x^2 + y^2 - 6x + 4y = 3 \]For the x variable:
  • Coefficient of x is -6
  • Divide by 2, we get -3
  • Square it, which results in 9. Add and subtract 9 in the equation.
Next, for the y variable:
  • Coefficient of y is 4
  • Divide by 2, you get 2
  • Square it to get 4. Add and subtract 4 in the equation.
This results in: \[ x^2 - 6x + 9 + y^2 + 4y + 4 = 16 \]We're now ready to rewrite the equation into its standard form.
Circle Center
Finding the center of a circle from its equation requires us to put it into standard form. The standard form of a circle's equation is \[ (x - h)^2 + (y - k)^2 = r^2 \] where \(h, k\) is the center, and \(r\) is the radius.
After completing the square, we get the equation: \[ (x - 3)^2 + (y + 2)^2 = 16 \] Here, we can directly identify the values:
  • The term \( (x - 3) \) indicates that \( h = 3 \).
  • The term \( (y + 2) \) indicates that \( k = -2 \).
Therefore, the center of the circle is at the point \( (3, -2) \). This shows how we use the completed square form to locate the circle's precise center point.
Circle Radius
Finding the radius of a circle from its equation is straightforward once it is in the standard form. Recall the standard form equation of a circle: \[ (x - h)^2 + (y - k)^2 = r^2 \].
In our example, the equation we derived is: \[ (x - 3)^2 + (y + 2)^2 = 16 \]. Here, \( r^2 = 16 \), which indicates:
  • To find the radius \( r \), we take the square root of 16.
  • The square root of 16 is 4.
So, the radius \( r \) of the circle is 4 units.
In summary, converting the given equation to the standard form not only reveals the center but also the radius of the circle, providing a complete description of the circle's geometry.

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Most popular questions from this chapter

Gigi's Cupcake Truck is about to open for business in a city of \(3,000,000\) people, traveling to several curbside locations in the city each day to sell cupcakes. The owners plan an advertising campaign that they think will induce \(30 \%\) of the people to buy their cupcakes. They estimate that if those people like the product, they will induce \(30 \% \cdot 30 \% \cdot 3,000,000\) more to buy the product, and those will induce \(30 \% \cdot 30 \% \cdot 30 \% \cdot 3,000,000\) and so on. In all, how many people will buy Gigi's cupcakes as a result of the advertising campaign? What percentage of the population is this? (PICTURE CANNOT COPY)

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