Problem 57 - 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. $$ x^{2}+y^{2}+8 x-2 y-8=0 $$

Step-by-Step Solution

Answer

The center of the circle is (-4, 1) and its radius is 5.

Step 1: Rearrange the equation grouping x's and y's together

Rearrange the equation of the circle such that all terms involving x are together, all terms involving y are together, and the constant term is on the other side, which gives \( x^{2} + 8 x + y^{2} - 2y = 8 \)

Step 2: Complete the square for x and y

For completing the square, you need to add and subtract the square of half the coefficient of x and y inside their respective brackets. This yields: \((x^{2} + 8x + 16) -(16) + (y^{2} - 2y + 1) - (1) = 8 \) Which simplifies to: \((x + 4)^{2} - 16 + (y - 1)^{2} - 1 = 8 \)

Step 3: Simplify the equation to a standard form

Simplify the equation to put it in a standard form. Here, we combine all the constant terms on the right-hand side of the equation, so this will give: \((x + 4)^{2} + (y - 1)^{2} = 25 \)

Step 4: Identify the center and radius

The equation of the circle in standard form is \( (x - a)^{2} + (y - b)^{2} = r^{2} \). Comparing this with our equation \((x + 4)^{2} + (y - 1)^{2} = 5^{2}\), you can see that a = -4, b = 1, and r = 5. Therefore, the center of the circle is (-4, 1) and the radius is 5.

Step 5: Graph the circle

Plot the center of the circle (-4,1) on the graph. Then, move 5 units (radius) away from the center in all directions and make the circle.

Key Concepts

Standard Form of Circle Equation

The standard form of a circle equation is a neat way to express circular shapes in coordinate geometry. It is crucial for identifying the center and radius of a circle easily. This form is written as:\[(x - a)^2 + (y - b)^2 = r^2\]Here,

• \((x - a)^2\) and \((y - b)^2\) are terms derived through a method called completing the square.
• \(a\) and \(b\) represent the coordinates of the circle's center.
• \(r\) is the radius, which tells us how far the circle stretches from its center.

The main goal when converting an equation into this form is to isolate these components, making it easier to visualize and graph later. This systematic rewriting is what enables us to find out other properties of the circle.

Center of a Circle

The center of a circle is a pivotal point, often symbolized by coordinates in a plane. In the formula \((x - a)^{2} + (y - b)^{2} = r^{2}\), the values

• \(a\), and
• \(b\)

represent the center's x-coordinates and y-coordinates respectively.
In our example, after rewriting the equation to standard form,

• the center lands at the point \((-4, 1)\).

This insight is critical because the entire structure of the circle revolves around this central point. It's where the distance of every point on the circle, which is the radius, originates. Knowing the center's location helps in sketching the circle on a coordinate plane and is the first step in finding where to place the circle graphically.

Radius of a Circle

A circle's radius is the constant distance from its center to any point along its edge. In the equation \[(x - a)^2 + (y - b)^2 = r^2\]\(r\) denotes the radius. It is determined by simplifying the terms linked with the standard form. In this instance, through completing the square and rearranging, we found

• \(r = 5\)

This radius indicates how far the circle expands from its center, \((-4, 1)\) in our case, in every direction. Understanding the radius size is fundamental not only in graphing but also in solving real-world problems involving circular paths or boundaries.

Graphing Circles

Graphing a circle involves translating the equation of a circle on a coordinate plane visually. After determining the standard form, center, and radius:

• Begin by plotting the center of the circle, which is \((-4, 1)\) in our example.
• Then, use the radius, here it is 5, to draw the curve. Move \(5\) units in each direction (up, down, left, and right) from the center to ensure accurate placement.
• Finally, sketch the circle by connecting these boundary points smoothly in a round fashion.

Graphing provides a concrete view of how the circle sits within the plane. It helps to understand geometrical relationships better and aids in solving further problems related to the circle's geometric space.