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Chapter 12: Problem 43

Rewrite each equation in the standard form for the equation of a circle, and identify its center and radius. $$x^{2}-3 x+y^{2}-y=1$$

Short Answer

Expert verified
Center: \( \left( \frac{3}{2}, \frac{1}{2} \right) \), Radius: \( \sqrt{\frac{7}{2}} \)

Step by step solution

01

- Group x and y terms

Rewrite the equation by grouping the x terms and y terms together: \[x^2 - 3x + y^2 - y = 1\]
02

- Complete the square for x terms

To complete the square for the x terms, take the coefficient of \(x\) (which is -3), divide by 2, and then square it: \[\left( \frac{-3}{2} \right)^2 = \left(\frac{-3}{2}\right)^2 = \frac{9}{4}\]Add and subtract \(\frac{9}{4}\) inside the equation:\[x^2 - 3x + \frac{9}{4} + y^2 - y = 1 + \frac{9}{4}\]
03

- Complete the square for y terms

Similarly, to complete the square for the y terms, take the coefficient of \(y\) (which is -1), divide by 2, and then square it:\[\left(\frac{-1}{2}\right)^2 = \left(\frac{-1}{2}\right)^2 = \frac{1}{4}\]Add and subtract \(\frac{1}{4}\) inside the equation:\[x^2 - 3x + \frac{9}{4} + y^2 - y + \frac{1}{4} = 1 + \frac{9}{4} + \frac{1}{4}\]
04

- Simplify and rearrange the equation

Combine and simplify the equation:\[\left( x - \frac{3}{2} \right)^2 + \left( y - \frac{1}{2} \right)^2 = \frac{14}{4}\]Which simplifies to:\[\left( x - \frac{3}{2} \right)^2 + \left( y - \frac{1}{2} \right)^2 = \frac{7}{2}\]
05

- Identify center and radius

The equation is now in the standard form of a circle, \[(x-h)^2 + (y-k)^2 = r^2\]where the center \((h, k)\) is \( \left( \frac{3}{2}, \frac{1}{2} \right) \)and the radius r is \( r = \sqrt{\frac{7}{2}} \)

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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 technique helps in transforming equations to make them easier to solve or analyze. For a quadratic term like \(x^2 - 3x\), you complete the square by following these steps:
  • Take the coefficient of x (which is -3), divide by 2 to get \(-\frac{3}{2}\)
  • Square the result to get \(\left(\frac{-3}{2}\right)^2 = \frac{9}{4}\)

Add and subtract this value inside the equation helps to create a perfect square trinomial. This transformation makes it possible to re-write the equation in a simpler and more recognizable form.
Now do the same with the \(y\) terms as seen in the solution.
center of a circle
The center of a circle in the standard form of the circle's equation \((x-h)^2 + (y-k)^2 = r^2 \) is the point \( (h, k) \). This form makes it straightforward to identify the circle's center by simply looking at the values \(h\) and \(k\).
Using our example:
\((x-\frac{3}{2})^2 + (y-\frac{1}{2})^2 = \frac{7}{2} \),
we can see that the coordinates of the center are \left( \frac{3}{2}, \frac{1}{2} \right)\.
radius of a circle
The radius of a circle is the distance from the center to any point on the circle. In the standard form equation of a circle, \((x-h)^2 + (y-k)^2 = r^2\), \( r^2 \) represents the square of the circle's radius. To find the radius, take the square root of the constant term on the right side of the equation.
In our example:
\((x - \frac{3}{2})^2 + (y - \frac{1}{2})^2 = \frac{7}{2} \),
we identify \r^2 = \frac{7}{2}\.
Therefore, the radius \ r \ is \sqrt{\frac{7}{2}}\.
algebraic manipulation
Algebraic manipulation is the process of rearranging and simplifying equations using various algebraic techniques. When solving for the standard form of a circle equation, this involves:
  • Grouping terms with similar variables (x terms together and y terms together)
  • Using methods like completing the square to transform parts of the equation into recognizable formats (perfect square trinomials in this context)
  • Simplifying fractions and combining terms to reach the desired form

Mastering algebraic manipulation techniques is crucial for solving a wide range of mathematical problems, including converting equations to standard forms like the circle equation we discussed here.

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Most popular questions from this chapter

Graph both equations of each system on the same coordinate axes. Solve the system by elimination of variables to find all points of intersection of the graphs. $$\begin{aligned}&(x-2)^{2}+(y+3)^{2}=4\\\&y=x-3\end{aligned}$$

Sketch the graph of each parabola. $$ y=(x+3)^{2}-1 $$

Graph each hyperbola and write the equations of its asymptotes. $$ y^{2}-x^{2}=1 $$

Solve each problem. Find all points of intersection of the parabolas \(y=x^{2}\) and \(x=y^{2}\)

Sketch the graph of each ellipse. $$ \frac{x^{2}}{9}+\frac{y^{2}}{16}=1 $$
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