Problem 70 - College Algebra

Question

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

Step-by-Step Solution

Answer

Center of the circle is (-5, 2) and radius is 7 units. Plot these on a graph for the solution.

Step 1: Rewrite Equation in Standard Form

The standard form of the equation of a circle is \( (x - h)^{2} + (y - k)^{2} = r^{2} \), where \( (h,k) \) are the coordinates of the center of the circle and \( r \) is the radius. Let's start by rewriting the given equation, \( x^{2} + 10x + y^{2} - 4y - 20 = 0 \), by completing the square for both x and y terms.

Step 2: Complete the Square and Find the Center

We complete the square by adding and subtracting the square of half the coefficient of x inside the brackets. Likewise, we do the same for the y terms. The equation becomes \( (x + 5)^{2} -25 + (y - 2)^{2} -4 - 20 = 0 \). Simplifying further we get \( (x + 5)^{2} + (y - 2)^{2} = 49 \), which is now in standard form. Hence, we get the center of the circle as (-5, 2).

Step 3: Identify the Radius

Remembering that the standard form is \( (x - h)^{2} + (y - k)^{2} = r^{2} \), it can be seen that \( r^{2} = 49 \). Hence, the radius \( r = \sqrt{49} = 7 \). This gives us all the information we need to graph this circle.

Step 4: Graph the Circle Using the Center and Radius

With a graphing utility, draw a circle with center (-5, 2) and a radius of 7 units. The x and y values will range from -12 to 2 and -5 to 9 respectively to fully capture the circle in the graph.