Problem 59 - 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 standard form of the equation is \((x - 1)^2 + y^{2} = 16\). The center of the circle is at point (1,0) and its radius is 4 units.

Step 1: Complete the Square

First, the given equation \(x^{2}-2 x+y^{2}-15=0\) needs to be rearranged into the form \(x^{2}-2 x + c + y^{2} = 15 + c\). The value 'c' will make the left hand side a perfect square. In this case, considering the coefficient of x, c should be \((\frac{-2}{2})^2 = 1\). So we add 1 in both sides of the equation.

Step 2: Write Equation in Standard Form

With the added 1, the equation can be rewritten as \((x - 1)^2 + y^{2} = 16\). This is the standard form.

Step 3: Identify the Center and the Radius

In the standard form, the center of the circle is (h,k) where h and k are the coefficients of x and y respectively. Thus the center of the circle is (1, 0). Here, 16 is the square of the radius. So, the radius of the circle is \(\sqrt{16} = 4\)

Step 4: Graph the Equation

Plot the center of the circle at point (1,0) on a graph. Then, draw a circle with radius 4 units around the center point.

Key Concepts

The Standard Form of a Circle

The standard form of a circle equation is a crucial concept in geometry. It simplifies the representation and analysis of a circle's properties. The standard form is given by:\[(x - h)^2 + (y - k)^2 = r^2\]where:

• \(h\) and \(k\) are the coordinates of the center of the circle.
• \(r\) is the radius of the circle.

This form allows us to quickly identify the center and radius of the circle at a glance. It's essentially a restatement of the Pythagorean Theorem for every point on the circle. Every point \((x, y)\) that satisfies this equation is exactly \(r\) units away from the center \((h, k)\).
This makes the standard form very useful for graphing circles, as both the location of the circle and its size are immediately clear.

Understanding the Center of a Circle

The center of a circle is a significant point that defines the exact middle of the circle. In the standard form equation \((x - h)^2 + (y - k)^2 = r^2\), the terms \(h\) and \(k\) are used to identify this center.

• The coordinate \(h\) is the x-coordinate of the center.
• The coordinate \(k\) is the y-coordinate of the center.

Understanding the center is crucial for positioning the circle on the coordinate plane.
In our example, where the equation becomes \((x - 1)^2 + y^2 = 16\), the center of the circle is \((1, 0)\). This means that from point (1,0), the circle is drawn with equal radius in all directions.

Exploring the Radius of a Circle

The radius of a circle is the distance from its center to any point on its boundary. It's a key measure defining the size of the circle. In the standard equation format \((x-h)^2+(y-k)^2=r^2\), the term \(r\) represents the radius.

• \(r^2\) is the radius squared.
• Therefore, to find the radius \(r\), you take the square root of the value on the right side of the equation.

In our given example, after completing the square and rearranging the equation to \((x - 1)^2 + y^2 = 16\), we can see that \(r^2 = 16\).
This means, the radius \(r\) is \(\sqrt{16} = 4\).
Thus, every point on the circle is exactly 4 units away from the center, (1,0). This consistency in distance helps to perfectly round the circle's shape.