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Chapter 13: Problem 14

Find the center and radius of each circle. Then graph the circle. $$ x^{2}+y^{2}=36 $$

Short Answer

Expert verified
The center is (0,0) and the radius is 6.

Step by step solution

01

- Recognize Circle Equation Form

Recognize that the equation given is in the standard form of a circle equation. The general form is: \[ (x - h)^2 + (y - k)^2 = r^2 \] where - \(h,k\) is the center of the circle - \(r\) is the radius.
02

- Compare and Identify Center

Compare the given equation \[ x^2 + y^2 = 36 \] with the standard form. Notice that \[ (x - 0)^2 + (y - 0)^2 = 36 \] implies the center \(h,k\) is \(0,0\).
03

- Determine the Radius

Set the given radius squared term equal to \[ r^2 = 36 \] and solve for \(r\): \[ r = \sqrt{36} = 6 \]
04

- Graph the Circle

Graph the circle using the center \(0,0\) and radius \(6\). From the center, measure 6 units in all directions (up, down, left, right) to plot points and then draw the circle connecting these points.

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Key Concepts

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

standard form of a circle
The standard form of a circle's equation helps us easily identify its key features. The standard form is: \[ (x - h)^2 + (y - k)^2 = r^2 \]Here,
  • \(h\) and \(k\) represent the coordinates of the circle's center.
  • \(r\) is the radius of the circle. The term \(r^2\) indicates the square of the radius.
By comparing your equation to this form, you can directly extract the circle's center and radius. Understanding this form will make solving circle equations straightforward and intuitive.
center of a circle
The center of a circle is a crucial point that defines its position. From the standard form of a circle's equation: \[ (x - h)^2 + (y - k)^2 = r^2 \]The center of the circle is at \((h, k)\). For the given equation \[ x^2 + y^2 = 36 \]we can rewrite it as \[ (x - 0)^2 + (y - 0)^2 = 36 \]This shows that the circle is centered at the origin, \((0,0)\).Picking out the center from the equation helps in graphing and further calculations.
radius of a circle
The radius of a circle is the distance from its center to any point on its circumference. From the standard form: \[ (x - h)^2 + (y - k)^2 = r^2 \]we know that the term on the right side of the equation is \(r^2\). For the given equation \[ x^2 + y^2 = 36 \]we have \[ r^2 = 36 \]To find the radius, we take the square root of 36: \( r = \[ \sqrt{36} = 6\] \). Hence, for the given circle, the radius is 6 units. This tells us how far the circle extends from its center.
graphing circles
Graphing a circle involves plotting its center and radius on a coordinate plane. Here's a clear step-by-step method to graph the given circle \[ x^2 + y^2 = 36 \]:
  • Identify the center: For this equation, the center is at \((0,0)\).
  • Identify the radius: The radius is 6 units.
  • From the center, move 6 units in all four primary directions (up, down, left, right) to determine key points that lie on the circle.
    • Up: \((0,6)\)
    • Down: \((0,-6)\)
    • Left: \((-6,0)\)
    • Right: \((6,0)\)
  • Connect these points smoothly to form a circle. These steps give you a clear and accurate graph of the circle.
Graphing helps visualize the circle's size and position on the plane.

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Most popular questions from this chapter

Solve each system using the elimination method or a combination of the elimination and substitution methods. $$ \begin{array}{l} x^{2}+x y-y^{2}=11 \\ x^{2}-y^{2}=8 \end{array} $$

Graph each system of inequalities. \(x^{2}+y^{2} \geq 4\) \(x+y \leq 5\) \(x \geq 0\) \(y \geq 0\)

Graph each system of inequalities. \(y-2\) \(x+y<3\) \(3 x-2 y>-6\)

Graph each generalized square root function. Give the domain and range. $$ \frac{y}{2}=\sqrt{1+\frac{x^{2}}{4}} $$

Graph each generalized square root function. Give the domain and range. $$ f(x)=-\sqrt{36-x^{2}} $$
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