Problem 54 - Precalculus 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 circle's standard form equation is \( (x+4)^2 + (y+2)^2 = 20 \). The center is at (-4,-2) and the radius is 2√5.

Step 1: Rewrite the equation to identify specific terms

Rearrange the terms of the equation to be grouped together: \( (x^2 + 8x) + (y^2 + 4y) + 16 = 0 \). So, the equation can be prepared for completing the square.

Step 2: Complete the square for each variable

Complete the square for each variable by adding the square of half the coefficient of x and the square of half the coefficient of y to both sides of the equation. This results in \( (x^2 + 8x +16) + (y^2 + 4y + 4) = 16 +4 . The equation now can be rewritten as \( (x+4)^2 + (y+2)^2 = 20.\)

Step 3: Identify the center and radius of the circle

From the standard form of the circle equation \( (x-h)^2 + (y-k)^2 = r^2 \), we can now identify the center (h,k) and radius r. The center of the circle is at (-4,-2) and the radius of the circle is \( \sqrt{20} \) which simplifies to 2√5.

Step 4: Graph the circle

On the coordinate plane, plot the center point at (-4,-2). Draw a circle with radius 2√5, using the center point as reference.

Key Concepts

Standard Form of Circle

The standard form of a circle's equation is essential for understanding and graphing circles. It is expressed as \((x-h)^2 + (y-k)^2 = r^2\). This formula is pivotal because it directly communicates the circle's center and its radius without further calculation.

• **\(h\) and \(k\)** represent the x and y coordinates of the circle's center.
• **\(r\)** is the radius of the circle, squared in the equation to maintain consistency with other quadratic equations.

With this form, recognizing the essential traits of a circle becomes straightforward, making it easier to sketch and understand their properties. Completing the square is a fundamental step considered here to transform a general quadratic equation into this standard form, as seen in the exercise above. By rearranging and completing the square on the given equation, we transform it into \((x+4)^2 + (y+2)^2 = 20\), hence evidencing the circle's configuration.

Center and Radius of Circle

Identifying the center and radius from the standard form of a circle is simple once you are familiar with the formula \((x-h)^2 + (y-k)^2 = r^2\).

• To identify the **center** \( (h,k) \), observe the values of \(h\) and \(k\), which are found inside the parenthesis with \(x\) and \(y\). In our example, \((x+4)^2 + (y+2)^2\) translates to a center at \((-4, -2)\).
• The **radius** \(r\), is straightforward as well – since the equation provides \(r^2\), you will need to take the square root of that term. For instance, \(r^2\) is 20 in our equation, which means \(r\) is \(\sqrt{20}\) or simplified, \(2\sqrt{5}\).

Once the center and radius are determined, the circle's geometric translation becomes highly intuitive.

Graphing Circles

Graphing circles from their equation in standard form is the culmination of understanding both their algebraic and geometric properties.
Here’s how to graph a circle effectively:

• **Plot the Center:** Start by plotting the center of the circle on a coordinate plane. From our exercise, the center is \((-4,-2)\). This is your reference point.
• **Draw the Circle Using the Radius:** Use the radius to measure out from the center. For a radius of \(2\sqrt{5}\), calculate its approximate value, which gives nearly 4.47. From the center, measure this distance in all directions to draw the circle.

Understanding the center and radius arms you with everything required to physically represent the circle. Combining algebraic manipulation with graphical interpretation solidifies knowledge of circle equations.