Problem 60 - College Algebra Essentials

Question

complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$ x^{2}+y^{2}-6 y-7=0 $$

Step-by-Step Solution

Answer

The equation of the circle in standard form is \((x-0)^{2}+(y-3)^{2}=2\). The center of the circle is at the point \((0,3)\) and the radius is \(\sqrt{2}\) or approximately 1.414.

Step 1: Grouping the Variables

Rearrange the equation to group the \(x\) and \(y\) terms together and move the constant to the right side of the equation: \(x^{2}+(y^{2}-6y)=-7\)

Step 2: Complete the Square

To complete the square for the \(y\) terms, add the square of half of the \(y\)'s coefficient to both sides of the equation. Half of -6 is -3, and \(-3^{2}=9\). So, the equation becomes: \(x^{2}+y^{2}-6y+9=-7+9\) which simplifies to \(x^{2}+(y-3)^{2}=2\)

Step 3: Write in Standard Form

Now the equation is nearly in the standard form, we rewrite it as \((x-0)^{2}+(y-3)^{2}=2\)

Step 4: Identify the Center and the Radius

In the standard form, the center is \((h,k)\) and radius is \(r\). Therefore for our equation \((x-0)^{2}+(y-3)^{2}=2\), the center is \((0,3)\) and the radius \(r\) is the square root of 2, which is approximately 1.414

Step 5: Graphing the Circle

The circle can be graphed using the center and radius. Plot the center point at \((0,3)\) on the graph. From the center, measure out a distance of approximately 1.414 (the radius) in all directions to draw the circle

Key Concepts

Circle Equation

In algebra and geometry, the circle equation is a fundamental concept used to describe all the points that are equidistant from a given center point. This idea can be visualized by imagining a point in the middle (the center), and the radius is the constant distance from this center point. For any circle, every point lies at the same distance from this center.

The general form of a circle's equation is:

• \( x^2 + y^2 + Dx + Ey + F = 0 \)

Sometimes, equations of circles may not appear to be in this format at first glance, as they could be scattered with terms. However, by manipulating such equations—through techniques like completing the square—we can transform them into a more recognizable form. This helps identify the center and radius more easily, leading us to the next topic: the standard form of a circle.

Standard Form of a Circle

The standard form of a circle's equation makes it very straightforward to figure out the circle's geometric properties, such as its center and radius. The advantage of using this form lies in its simplicity and clarity. The equation appears as:

• \( (x-h)^2 + (y-k)^2 = r^2 \)

Here,

• \((h, k)\) represents the center of the circle.
• \(r\) is the radius of the circle.

By completing the square from the general form, a circle's equation can be converted to this standard form. Let's see how this works in the example provided:

• Begin by rearranging and grouping similar terms—something already done in the problem statement.
• Next, adjust these terms so they become perfect squares using the completing the square method. This is just a way to create perfect square trinomials which are easy to factor.

From our example, the terms \(x^2 + (y-3)^2 = 2\) provide us clear information. The center is \((0,3)\) and the radius \(r\) is \(\sqrt{2}\).

Graphing Circles

Having crafted the circle's equation into its standard form, graphing becomes a much simpler task. You begin by identifying the circle's center and radius from the provided standard form equation. Each of these components provides crucial information required for an accurate graph.

To graph the circle:

• Plot the center of the circle on the coordinate plane. This is your fixed point.
• Using the radius, measure out equal distance from the center in all directions. This establishes the boundary of the circle.

For example, the standard form \((x-0)^2 + (y-3)^2 = 2\) shows us that the center is at \((0, 3)\). The radius being \(\sqrt{2}\) means you measure about 1.414 units out in each direction.

Armed with this form and these steps, you can draw a perfect circle. Understanding this helps in visualizing geometric relationships and connections in algebra.