Problem 55 - 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. $$x^{2}-2 x+y^{2}-15=0$$

Step-by-Step Solution

Answer

The center of the circle is at (1,0) and the radius is 4 units.

Step 1: Rearrange Equation

Rearrange the given equation to group the x terms and the y terms together: \( x^{2} - 2x + y^{2} = 15 \)

Step 2: Complete the Square for x

To complete the square for the x terms, find the square of half the coefficient of x. In this case, half of -2 is -1, and squaring it gives 1. Add this to both sides of the equation: \( x^{2} - 2x + 1 + y^{2} = 15 + 1 \). This makes the equation: \( (x-1)^{2} + y^{2} = 16 \)

Step 3: Identify Circle Parameters

Now, the equation is in standard form for a circle. Identify the center and radius of the circle. The center of the circle can be found at the point (1,0) and the radius is the square root of 16, i.e 4.

Step 4: Graph the Circle

Plot the center of the circle on the graph. Then, from the center, draw a circle with a radius of 4 units.