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Magazine Chapter 10: Problem 64
Find the center and radius of each circle. $$x^{2}+5 x+y^{2}+3 y=\frac{1}{2}$$
Short Answer
Expert verified
Center: \( (-\frac{5}{2}, -\frac{3}{2}) \), Radius: 3
Step by step solution
01
Rewrite the equation
First, rewrite the given equation in standard form. The given equation is\[ x^2 + 5x + y^2 + 3y = \frac{1}{2} \]
02
Move the constant term to the right side
Move the constant term to the right side of the equation:\[ x^2 + 5x + y^2 + 3y = \frac{1}{2} \] Now subtract \( \frac{1}{2} \) from both sides:\[ x^2 + 5x + y^2 + 3y - \frac{1}{2} = 0 \]
03
Complete the square for the x-terms
To complete the square for the \( x \)-terms, take half of the coefficient of \( x \) (which is 5), square it, and add and subtract it:\[ x^2 + 5x = (x^2 + 5x + \frac{25}{4}) - \frac{25}{4} = (x + \frac{5}{2})^2 - \frac{25}{4} \]
04
Complete the square for the y-terms
To complete the square for the \( y \)-terms, take half of the coefficient of \( y \) (which is 3), square it, and add and subtract it:\[ y^2 + 3y = (y^2 + 3y + \frac{9}{4}) - \frac{9}{4} = (y + \frac{3}{2})^2 - \frac{9}{4} \]
05
Combine completed squares
Combine the completed squares and constant terms on the right side:\[ (x + \frac{5}{2})^2 - \frac{25}{4} + (y + \frac{3}{2})^2 - \frac{9}{4} = \frac{1}{2} \]
06
Simplify the equation
Combine all constant terms and move them to the right side:\[ (x + \frac{5}{2})^2 + (y + \frac{3}{2})^2 = \frac{1}{2} + \frac{25}{4} + \frac{9}{4} \] Simplify:\[ (x + \frac{5}{2})^2 + (y + \frac{3}{2})^2 = \frac{2}{4} + \frac{25}{4} + \frac{9}{4} = \frac{36}{4} = 9 \]
07
Identify the center and radius
The final equation in standard form is:\[ (x + \frac{5}{2})^2 + (y + \frac{3}{2})^2 = 9 \] This represents a circle with center at \((-\frac{5}{2}, -\frac{3}{2})\) and radius \( r = \sqrt{9} = 3 \)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
completing the square
The process of completing the square is a method used to rewrite a quadratic equation into a perfect square trinomial. This is very useful in transforming the equation of a circle into its standard form. For example, consider the equation provided:
\[ x^2 + 5x + y^2 + 3y = \frac{1}{2} \],
we start by isolating the linear terms involving x and y:
\[ x^2 + 5x \text{ and } y^2 + 3y \].
\[ x^2 + 5x + y^2 + 3y = \frac{1}{2} \],
we start by isolating the linear terms involving x and y:
\[ x^2 + 5x \text{ and } y^2 + 3y \].
- First, you take half of the coefficient of x (which is 5), square it, and both add and subtract this value inside the equation: (5/2)虏 = 25/4.
- Next, you do the same for the y terms. Take half of 3, square it, and add and subtract: (3/2)虏 = 9/4.
\[ x^2 + 5x = (x + \frac{5}{2})^2 - \frac{25}{4} \].
\[ y^2 + 3y = (y + \frac{3}{2})^2 - \frac{9}{4} \].
center of a circle
The center of a circle in the equation \[ (x - h)^2 + (y - k)^2 = r^2 \] is at the point \( (h, k) \). After completing the square for both x and y terms, the initial circle equation can be transformed as follows:
\[ (x + \frac{5}{2})^2 + (y + \frac{3}{2})^2 = 9 \].
Notice that we now have an equation in the form \[ (x - (-\frac{5}{2}))^2 + (y - (-\frac{3}{2}))^2 = 9 \], indicating that the center of the circle is at:
\[ (x + \frac{5}{2})^2 + (y + \frac{3}{2})^2 = 9 \].
Notice that we now have an equation in the form \[ (x - (-\frac{5}{2}))^2 + (y - (-\frac{3}{2}))^2 = 9 \], indicating that the center of the circle is at:
- \((h, k) = (-\frac{5}{2}, -\frac{3}{2})\).
radius of a circle
The radius of a circle in the standard form equation \[ (x - h)^2 + (y - k)^2 = r^2 \] is given by the square root of the right-hand side constant. In the exercise, after completing the square and simplifying all terms, the final equation becomes:
\[ (x + \frac{5}{2})^2 + (y + \frac{3}{2})^2 = 9 \].
With this equation, we see that the right-hand side is 9, making the radius:
\[ r = \sqrt{9} = 3 \].
The radius tells us how far any point on the circle is from the center. This distance is consistent for every point around the circle.
\[ (x + \frac{5}{2})^2 + (y + \frac{3}{2})^2 = 9 \].
With this equation, we see that the right-hand side is 9, making the radius:
\[ r = \sqrt{9} = 3 \].
The radius tells us how far any point on the circle is from the center. This distance is consistent for every point around the circle.
