Problem 85 - Precalculus Essentials

Question

Use a graphing utility to graph each circle whose equation is given. Use a square setting for the viewing window. $$x^{2}+10 x+y^{2}-4 y-20=0$$

Step-by-Step Solution

Answer

The center of the circle is (-5, 2) and the radius is 5. Draw the circle using this information.

Step 1: Identify the center and radius

The given equation can be rewritten as: \(x^{2}+10x+y^{2}-4y=-20\). It can be observed that \(D=10\) and \(E=-4\) from which the center coordinates (-D/2, -E/2) can be calculated to be (-5, 2). Next, F is equal to 20 from which the radius can be calculated as: \(\sqrt{(10/2)^2+(-4/2)^2-20}\) which equals \(\sqrt{25}\) or 5.

Step 2: Graph the center

On a graphing utility, navigate to the point (-5, 2) which is the center of the circle. Mark this point.

Step 3: Draw the circle

Using the center as the starting point, draw a circle with a radius of 5. Make sure to adjust the viewing window to a square setting to properly visualize the circle.

Key Concepts

Circle Equation

To graph a circle, you first need to understand its equation. The standard form of a circle's equation is \((x-h)^2 + (y-k)^2 = r^2\), where \((h,k)\) represents the center of the circle and \(r\) is the radius.
In the given equation, \(x^2 + 10x + y^2 - 4y - 20 = 0\), identifying the circle's attributes involves rewriting it into the standard form.
This requires completing the square for both \(x\) and \(y\) terms. Once rewritten, the equation reveals the center's coordinates and the radius length, which are essential for graphing.

• To complete the square for \(x\): take half of 10 (which is 5), square it (25), and balance the equation accordingly.
• For \(y\), take half of -4 (which is -2), square it (4), and adjust the equation accordingly.

By these actions, the equation becomes simpler and enables you to easily identify important circle values.

Graphing Utility

Utilizing a graphing utility makes plotting circles straightforward and accurate. These tools allow for adjustments such as zoom level and window settings that are crucial for visual representation.
When using a graphing utility, ensure the viewing window is square. This setting prevents distortion, ensuring that circles appear round instead of elliptical.

• Adjust the window to have equal units on the x and y axes by selecting the square option.
• Input the adjusted standard form of the circle's equation into the utility.

Graphing utilities provide visual feedback, helping you see and correct any errors directly, which can be especially helpful if your equation isn’t quite right. Such utilities are powerful tools for quickly checking your work and gaining understanding.

Center and Radius

The center and radius are key elements in a circle's graph. In our example, the center is calculated first by completing the square. With the given equation, the terms \(D = 10\) and \(E = -4\) help find the center at \((-\frac{D}{2}, -\frac{E}{2})\) or \((-5, 2)\). This point needs to be marked on the graph.
Once the center is plotted, calculating the radius involves simplifying \(\sqrt{(\frac{D}{2})^2 + (\frac{E}{2})^2 - F}\), where \(F\) is extracted from the constant term of the rewritten equation.

• In this case, \((-\frac{10}{2})^2 + (-\frac{-4}{2})^2 - 20 = 25\), hence the radius, \(r = \sqrt{25}\) or 5.

With these values on hand, drawing the circle becomes straightforward by centering it on \((-5, 2)\) with a radius of 5 units.