Problem 54 - 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. $$ x^{2}+y^{2}+8 x+4 y+16=0 $$

Step-by-Step Solution

Answer

The center of the circle is (-4, -2) and the radius is 2.

Step 1: Grouping

Regroup the equation by x's and y's: \(x^{2}+8 x+y^{2}+4 y+16=0\) into \(x^{2}+8x + y^{2}+4y = -16\)

Step 2: Completing the square for x

To complete the square for the x terms, take half the coefficient of x, square it and add it to both sides. This results in \(x^{2}+8x+16 + y^{2}+4y = -16+16\). Now, the x part can be written as a square: \((x+4)^2 + y^{2}+4y = 0\)

Step 3: Completing the square for y

Repeat step 2 for the y terms. Half of 4 is 2, and 2^2 is 4. Add it to both sides to get \((x+4)^2 + y^{2}+4y +4 = 4\). Now, the y part can be written as a square: \((x+4)^2 + (y+2)^2 = 4\)

Step 4: Finding the center and radius

From the standard form, we can infer the center and the radius of the circle. The center is given by the point (-h, -k), which in this case is \(-4,-2\), and the radius is the square root of the right side of the equation, giving us \(r=2\).

Step 5: Graphing the equation

Using the center, that's (-4, -2), as a starting point on the graph and r = 2 as the radius draw a circle.

Key Concepts

Standard Form of a Circle

Understanding the standard form of a circle's equation is foundational to studying circle graphing and its properties. The general standard form is written as \( (x-h)^2 + (y-k)^2 = r^2 \), where \(h, k\) represents the center of the circle, and \(r\) is the radius.

In simpler terms, this equation tells us that any point \( (x, y) \) that lies on the circle is exactly \(r\) units away from the center \( (h, k) \). This distance is consistent for all points on the circle, which is why it forms a perfect loop. Transitioning from the expanded form of a circle's equation to its standard form often involves completing the square, a technique that facilitates the identification of the circle’s center and radius.

For instance, to write \( x^{2}+y^{2}+8x+4y+16=0 \) in standard form, completing the square helps to restructure it to \( (x+4)^2 + (y+2)^2 = 4 \), which now clearly indicates the circle's radius and center.

Circle Graphing

Graphing a circle involves plotting it on a coordinate plane with a clear understanding of the standard equation's components. Once the standard form equation is known, the graphing process includes identifying the center point and then accurately drawing a circle that extends the radius' length in every direction from the center.

Exercise improvement advice suggests to first plot the center \( (h, k) \). Using the example \( (x+4)^2 + (y+2)^2 = 4 \), we would plot the point \( (-4, -2) \) on a coordinate grid. Next, you measure out the distance of the radius from this point. Given \(r=2\), from the center, measure two units in each cardinal direction (up, down, left, right) and mark these points. Then, carefully draw a curve through these points to complete the circle, ensuring uniformity in shape.

To assist students in visualizing the circle, sketching additional points equidistant from the center often helps before drawing the circumference. Graphing calculators or software can also be used for a more precise representation.

Circle Center and Radius

Identifying the circle's center and radius is pivotal for graphing and understanding its geometry. We derive these directly from the standard form of the equation \( (x-h)^2 + (y-k)^2 = r^2 \), as aforementioned.

The center \( (h, k) \) is simply the coordinate which is opposed in sign to the \(x\) and \(y\) in the binomial squares. Reading the example equation \( (x+4)^2 + (y+2)^2 = 4 \) reveals the center to be \( (-4, -2) \). Understanding that the radius is the constant distance from the center to any point on the circle's edge leads us to find that the radius \(r\) is the square root of the constant term on the equation's right side, resulting in \(r=2\) for our example.

Knowing the center and radius allows for a complete understanding of the circle’s size and position on the coordinate plane. It enables students to tackle exercises involving circles systematically, whether graphing, finding an equation, or solving for geometric properties within or outside the circle.