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Chapter 12: Problem 36

Write the center-radius form of the circle with the given equation. Give the center and radius, and graph the circle. $$x^{2}+y^{2}+8 x+2 y-8=0$$

Short Answer

Expert verified
(x+4)^2 + (y+1)^2 = 25; Center: (-4, -1); Radius: 5

Step by step solution

01

Rewrite the Equation

Rewrite the given equation in the form of a standard circle equation by grouping the x and y terms together: d:x^{2}+8x + y^{2}+2y - 8 = 0.
02

Complete the Square for x Terms

To complete the square for the x terms, take half of the coefficient of x (which is 8), square it (to get 16), and add and subtract this value inside the equation: x^2 + 8x + 16 + y^2 + 2y - 8 - 16 = 0.
03

Complete the Square for y Terms

Similarly, complete the square for the y terms. Take half of the coefficient of y (which is 2), square it (to get 1), and add and subtract this value inside the equation: x^2 + 8x + 16 + y^2 + 2y + 1 - 8 - 16 - 1 = 0.
04

Simplify the Equation

Simplify the equation by combining like terms and moving constants to the other side of the equation: d(x+4)^2 + (y+1)^2 - 25 = 0 Rewriting, we get: d(x+4)^2 + (y+1)^2 = 25.
05

Identify the Center and Radius

From the simplified equation d(x+4)^2 + (y+1)^2 = 25, we can identify the center and radius of the circle: dCenter: (-4, -1) Radius: 5 (since 25 is 5^2).
06

Graph the Circle

Using the center (-4, -1) and radius 5, plot the center on a coordinate graph and draw a circle with radius 5 centered at this point.

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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 technique used to simplify quadratic equations. It turns a quadratic equation into a perfect square trinomial, which makes it easier to solve or analyze. Here's how to complete the square:
  • Focus on one variable at a time. For example, if you have terms involving x, like x虏 + bx, work with those first.
  • Take the coefficient of the linear term (in this case, b), divide it by 2, and square it. This number will be added and subtracted to complete the square.
  • For example, with x虏 + 8x, the coefficient of x is 8. Half of 8 is 4, and 4 squared is 16. So, you would add and subtract 16: x虏 + 8x + 16 - 16.
  • Repeat the same process for the y terms.
This technique makes it easy to convert a standard quadratic equation into something more manageable. In our exercise, we used this method to rewrite x虏 + 8x as (虫+4)虏 and y虏 + 2y as (测+1)虏. This helped simplify the equation to neatly fit the circle's center-radius form.
equation of a circle
The general equation of a circle in center-radius form is \((x-h)^2 + (y-k)^2 = r^2\). Here, (h, k) denotes the center of the circle, and r is the radius.
  • In the given exercise, our goal was to convert the provided equation into this standard form.
  • This makes it much easier to identify the circle's center and radius.
After completing the square for both x and y terms, our equation transformed into \((x+4)^2+(y+1)^2=25\).
  • By comparing this to the standard form \((x-h)^2 + (y-k)^2 = r^2\), we can see that the center or (h, k) is (-4, -1),
  • while the radius or r is the square root of 25, which is 5.
Identifying the center and radius from the equation is crucial for graphing the circle and further understanding its properties.
graphing circles
Graphing a circle becomes straightforward once you have the center and the radius. Here are some steps to follow:
  • First, plot the center of the circle on the coordinate plane. For our exercise, the center is at (-4, -1).
  • Next, use the radius to mark points around the center. Since our radius is 5, measure 5 units up, down, left, and right from the center.
  • Draw a smooth, round curve connecting these points, forming the circle.

Make sure your circle is symmetrical around the center.
Using these steps, you can visualize a circle and better understand its properties and position on the graph.

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Most popular questions from this chapter

Use graphical methods on the given constraints to find the indicated optimal value of the given objective function. \(\begin{aligned} 2 x+3 y & \leq 100 \\ 5 x+4 y & \leq 200 \\ x & \geq 10 \\\ y & \geq 20 \\ \text { Maximize } & x+3 y . \end{aligned}\)

Graph each ellipse. $$\frac{(x-2)^{2}}{16}+\frac{(y-1)^{2}}{9}=1$$

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Solve each system using the substitution method. \(y=3 x^{2}+6 x\) \(y=x^{2}-x-6\)
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