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Chapter 10: Problem 13

Find the center and the radius of the circle with the given equation. Then draw the graph. $$x^{2}+y^{2}-6 x-8 y+16=0$$

Short Answer

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

Step by step solution

01

Rearrange the equation

Start by rearranging the given circle equation into a standard circle format. The given equation is: \[x^{2} + y^{2} - 6x - 8y + 16 = 0\]Move the constant term to the right side of the equation: \[x^{2} + y^{2} - 6x - 8y = -16\]
02

Complete the square for x

To complete the square for the x-terms, take half of the coefficient of x, square it, and add it to both sides of the equation. The coefficient of x is -6, half of that is -3, and squaring -3 gives 9. Add and subtract 9 for completing the square: \[x^{2} - 6x + 9 + y^{2} - 8y = -16 + 9\]This simplifies to: \[(x - 3)^{2} + y^{2} - 8y = -7\]
03

Complete the square for y

Similarly, complete the square for the y-terms. The coefficient of y is -8, half of that is -4, and squaring -4 gives 16. Add and subtract 16: \[(x - 3)^{2} + y^{2} - 8y + 16 = -7 + 16\]This simplifies to: \[(x - 3)^{2} + (y - 4)^{2} = 9\]
04

Identify the center and radius

The equation is now in standard form \[(x - h)^{2} + (y - k)^{2} = r^{2}\]Comparing, the center \((h, k) = (3, 4)\)and the radius \(r = \sqrt{9} = 3\)
05

Draw the graph

To graph the circle, plot the center at the point (3, 4) on the coordinate plane. Then draw a circle with radius 3 units away from the center. Make sure that every point on the circle is exactly 3 units away from the center.

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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 transform a quadratic equation into a perfect square trinomial, making it easier to solve or graph. For a given quadratic form like \(ax^2 + bx + c\), you can complete the square as follows:
  • First, divide the coefficient of the linear term (the coefficient of x, i.e., b) by 2.
  • Then, square the result.
  • Add and subtract this squared term within the equation to transform it into a perfect square.
This process helps to rewrite the quadratic equation in a more manageable form. For example, when completing the square for the equation \(x^2 - 6x\), you take half of -6 (which is -3), square it to get 9, and then add and subtract 9: \(x^2 - 6x + 9 - 9\). This becomes \((x - 3)^2 - 9\).
This technique is crucial when rewriting the general equation of a circle to find its center and radius.
Standard Form of a Circle
The standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius. This form is achieved by completing the square for both the x and the y terms in the given generalized equation of a circle. To convert an equation like \(x^2 + y^2 - 6x - 8y + 16 = 0\) into standard form:
  • First, rearrange and complete the square for the x and y terms.
  • Move the constant term to the other side of the equation.
  • After completing the square for both variables, you will get the equation into the form \((x - h)^2 + (y - k)^2 = r^2\).
In the given example, we rearranged the terms and completed the square to eventually get \((x - 3)^2 + (y - 4)^2 = 9\), which represents a circle with center \( (3, 4)\) and radius 3.
Graphing Circles
Graphing a circle involves plotting points that are equidistant from the center. Once you have the circle's equation in standard form, you can easily identify the center and radius:
  • Start by plotting the center \((h, k)\) on a coordinate plane.
  • Next, use the radius \(r\) to mark points that are exactly \(r\) units away from the center in all directions.
  • Draw a smooth curve through these points to form the circle.
For example, with the equation \((x - 3)^2 + (y - 4)^2 = 9\), plot the center at (3, 4). Then draw points 3 units away from this center in all directions, ensuring that the distance remains consistent. This will give you a circle neatly centered at (3, 4) with a radius of 3 units.
Radius and Center of a Circle
The radius and center are the fundamental elements that define a circle. In the standard form equation \((x - h)^2 + (y - k)^2 = r^2\):
  • \((h, k)\) represents the coordinates of the circle's center.
  • The radius \(r\) is the distance from this center to any point on the circle.
Identifying the center and radius from a given equation allows us to understand the circle's position and size in the coordinate plane. For example, in the equation \((x - 3)^2 + (y - 4)^2 = 9\), the center is \((3, 4)\) and the radius is \(\frac{9}{2}\) or 3 units. This means every point on the circle is exactly 3 units away from the center (3, 4). Knowing these values helps in graphing and understanding the circle's properties effectively.

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