Problem 50 - College Algebra

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: Rearrange the equation

The first step is to arrange the equation by grouping the x terms together and the y terms together: \(x^2 + 8x + y^2 + 4y = -16\)

Step 2: Decide what number to add to complete the square

The formula to complete the square is \( (x + a)^2 = x^2 + 2ax + a^2 \). To complete the square for the x terms, we need to figure out what number a to use in the formula to get 8x as our middle term. To do this, we simply divide the coefficient of the x term in our equation by 2, which gives us 4. Applying the same approach for y terms we get 2. Therefore, we add 16 (which is \(4^2 \)) to both sides of the equation after the x terms, and add 4 (which is \(2^2\)) to both sides of the equation after the y terms.

Step 3: Rewrite the equation

Once we know what number to add to complete the square for the x and y terms, we can rewrite the equation: \((x^2 + 8x + 16) + (y^2 + 4y + 4) = -16 + 16 + 4\). This simplifies to \((x + 4)^2 + (y + 2)^2 = 4\)

Step 4: Determine center and radius

The standard form of the equation of a circle is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is its radius. Comparing this with our standard form equation, we inferred that \(h = -4\), \(k = -2\) and \(r = \sqrt{4} = 2\). Therefore, the center of the circle is (-4, -2) and its radius is 2.

Step 5: Graph the equation

To graph this circle, you plot the center point at (-4, -2), then draw a circle with a radius of 2. Make sure to label your graph thoroughly so anyone looking at it can understand how you got to your final graph.