Problem 21 - Algebra 2

Question

Find the center and radius of the circle with the given equation. Then graph the circle. $$ x^{2}+y^{2}=144 $$

Step-by-Step Solution

Answer

The center is \((0, 0)\) and the radius is 12.

Step 1: Identify the Standard Circle Equation Form

The equation of a circle in standard form is \((x-h)^{2}+(y-k)^{2}=r^{2}\), where \((h,k)\) is the center and \(r\) is the radius.

Step 2: Compare Given Equation to Standard Form

Given equation is \(x^{2} + y^{2} = 144\). Notice there are no linear terms for \(x\) or \(y\), which indicates the center \((h, k)\) is at the origin \((0, 0)\).

Step 3: Determine the Center of the Circle

Since the equation is \((x-0)^{2} + (y-0)^{2} = 144\), the center of the circle is \((0,0)\).

Step 4: Find the Radius

The equation \(x^{2} + y^{2} = 144\) matches the form \((x-h)^{2} + (y-k)^{2} = r^{2}\) with \(r^{2} = 144\). Therefore, take the square root of \(144\) to find \(r = 12\).

Step 5: Graph the Circle

To graph the circle, plot the center at \((0,0)\), and draw a circle with a radius of 12 units around this point. The circle should intersect the x and y axes at \((12,0)\), \((-12,0)\), \((0,12)\), and \((0,-12)\).

Key Concepts

Standard Circle Equation Form

The standard equation for a circle provides a simple way to describe all the points that make up a circle on a coordinate plane. It is given by

• \((x-h)^{2} + (y-k)^{2} = r^{2}\)

In this form:

• \(h\) and \(k\) represent the coordinates of the circle's center, respectively.
• \(r\) is the radius, signifying the distance from the center to any point on the circle.

This equation makes it easier to visualize and graph circles. Every point \((x, y)\) on the circle will satisfy this equation. The absence of linear terms (just \(x^2\) and \(y^2\) terms) suggests the circle's center is at the origin, making some problems straightforward.

Center of a Circle

Determining the center of a circle from its equation helps set up its position on a graph. For the equation

• \((x-h)^{2} + (y-k)^{2} = r^{2}\),

you can easily find

• the center by identifying the values of \(h\) and \(k\).

In many cases, like

• where the equation simplifies to \(x^{2} + y^{2} = r^{2}\),

the center

• is at the origin, \((0, 0)\).

On graph paper, place the center at this point, and it acts as a pivot for the circle.

Radius of a Circle

The radius of a circle is a critical measure, indicating how large the circle will be on a graph. From the standard circle equation

• \((x-h)^{2} + (y-k)^{2} = r^{2}\),

identifying the radius involves taking the square root of

• the constant on the right side of the equation, \(r^{2}\).

In the case of

• \(x^{2} + y^{2} = 144\),

by solving for \(r\) you find that

• \(r = 12\),

meaning the radius is 12 units long from the center to any point on the edge of the circle. This measure helps in drawing an accurate depiction of the circle when graphing.

Graphing a Circle

Graphing a circle involves plotting a perfect round shape that encompasses all points equidistant from a center. To graph,

• start with placing a point at the center of the circle, \((h, k)\),

which could, as an example,

• be the origin \((0,0)\) when \(h = 0\) and \(k = 0\).

Next, using the radius,

• draw a circle with radius \(r = 12\) around this center.

This forms a circle hitting key intercepts with the axes, like

• \((12, 0)\), \((-12, 0)\), \((0, 12)\), and \((0, -12)\).

These points help you confirm the size and correct placement of the circle on the graph. By following these steps, you ensure a neat and precise representation.