Problem 27 - Intermediate Algebra

Question

The graph of each equation is a circle. Find the center and the radius and then graph the circle. $$(x-5)^{2}+(y+2)^{2}=1$$

Step-by-Step Solution

Answer

Center: (5, -2), Radius: 1.

Step 1: Recognize the Equation of a Circle

The given equation \((x-5)^2 + (y+2)^2 = 1\) fits the standard form of a circle's equation \((x-h)^2 + (y-k)^2 = r^2\), where \((h,k)\) is the center of the circle and \(r\) is the radius.

Step 2: Identify the Center of the Circle

From the equation \((x-5)^2 + (y+2)^2 = 1\), we can see that it matches \((x-h)^2 + (y-k)^2\). Here, \(h=5\) and \(k=-2\), so the center of the circle is \((5, -2)\).

Step 3: Determine the Radius of the Circle

The right side of the equation \((x-5)^2 + (y+2)^2 = 1\) gives us \(r^2 = 1\). Taking the square root of both sides, we find \(r = 1\). Thus, the radius of the circle is 1.

Step 4: Graph the Circle

To graph the circle, plot the center at \((5, -2)\) on the coordinate plane. From the center, draw a circle with radius 1, ensuring all points on the circumference are equidistant (1 unit) from the center.

Key Concepts

Understanding the Center of a Circle

In the standard equation of a circle, \((x-h)^2 + (y-k)^2 = r^2\), identifying the center is crucial.The terms \(h\) and \(k\) represent the coordinates of the circle's center.When dealing with equations in this form, subtracting \(h\) and \(k\) from \(x\) and \(y\) reveals the location of this critical point.For example, in the equation \((x-5)^2 + (y+2)^2 = 1\), analyzing suggests that \(h = 5\) and \(k = -2\).This translates to the center being at the point \((5, -2)\).Remember:

• The center \((h,k)\) is a fixed point from which all points on the circle are equidistant.
• Changing the values of \(h\) and \(k\) moves the circle around the coordinate plane.

Understanding the center helps correctly place your circle when graphing.

Finding the Radius of a Circle

The radius of a circle is the distance from its center to any point on its edge.In the equation \((x-h)^2 + (y-k)^2 = r^2\), this distance is represented by \(r\).To determine the radius, look at the right side of the equation.It provides \(r^2\).For the equation \((x-5)^2 + (y+2)^2 = 1\), the right side shows \(r^2 = 1\).Taking the square root, we find \(r = 1\).Hence, the radius is 1.Remember:

• The radius is always a non-negative number, even when \(r^2\) is a perfect square.
• Knowing the radius allows you to gauge the size of the circle.

Accurately identifying the radius is crucial for proper graphing.

Graphing Circles on a Coordinate Plane

Graphing a circle involves accurately plotting its size and position on a coordinate plane.Start by finding the center, then use the radius to draw the circumference.To graph the equation \((x-5)^2 + (y+2)^2 = 1\):

• First, pinpoint the center at \((5,-2)\).This determines the circle's exact position.
• From this center, measure the radius of 1 unit to establish the circle's edge.
• Sketch the circle so that every point on the edge maintains a 1-unit distance from the center \((5,-2)\).

Ensure your circle is accurately centered, and check that its diameter stretches precisely twice the radius's length.With precision, you can visualize and analyze the circle's characteristics on the graph.