Problem 66 - Algebra and Trigonometry

Question

65–72 ? Show that the equation represents a circle, and find the center and radius of the circle. $$ x^{2}+y^{2}-2 x-2 y=2 $$

Step-by-Step Solution

Answer

The circle's center is at (1, 1) and its radius is 2.

Step 1: Rewrite the equation

Start by rewriting the given equation \(x^2 + y^2 - 2x - 2y = 2\). In this form, the equation doesn't immediately look like the standard equation of a circle, which is \((x - h)^2 + (y - k)^2 = r^2\). We're going to rearrange and complete the square to see the circle's equation more clearly.

Step 2: Group and organize terms

Group the equation terms: \(x^2 - 2x + y^2 - 2y = 2\). This grouping will help in the completion of the square for both the \(x\) terms and the \(y\) terms separately.

Step 3: Complete the square for x-terms

Take the \(x\)-terms, \(x^2 - 2x\). To complete the square, add and subtract \(1\) inside the equation. The complete square form is \((x - 1)^2 = x^2 - 2x + 1\), therefore replace \(x^2 - 2x\) with \((x - 1)^2 - 1\).

Step 4: Complete the square for y-terms

Use the same technique for the \(y\)-terms, \(y^2 - 2y\). Add and subtract \(1\), the complete square is \((y - 1)^2 = y^2 - 2y + 1\), so replace \(y^2 - 2y\) with \((y - 1)^2 - 1\).

Step 5: Adjust the equation

Insert the complete squares back into the equation: \((x - 1)^2 - 1 + (y - 1)^2 - 1 = 2\). Simplify this to \((x - 1)^2 + (y - 1)^2 = 4\).

Step 6: Identify the circle's properties

Now that the equation is in the standard form \((x - h)^2 + (y - k)^2 = r^2\), we can easily identify that \(h = 1\), \(k = 1\), and \(r^2 = 4\). Thus, the circle's center is \((1, 1)\) and the radius \(r\) is \(2\), since \(r = \sqrt{4}\).

Key Concepts

Equation of a circle

In geometry, the equation of a circle is a crucial concept when dealing directly with geometric figures in a coordinate plane. The standard form of a circle's equation is given by \((x - h)^2 + (y - k)^2 = r^2\).
Here,

• \( (h, k) \) represents the center of the circle, which are the coordinates where the circle's center is located.
• \( r \) is the circle's radius, the distance from the center to any point on the outer edge of the circle.

When a circle's equation is not in this standard form, employing techniques such as completing the square can aid in transforming it. This helps in easily identifying the circle's center and radius, guiding us effortlessly in its graphical representation.

Completing the square

Completing the square is a valuable mathematical technique used to transform a quadratic expression into a perfect square trinomial. This process simplifies solving and graphing. For instance, if you have a quadratic expression like \(x^2 - 2x\), completing the square involves forming it into a phrase \((x-a)^2\), where \(a\) is a number that makes this square complete.
The steps are fairly straightforward:

• Identify the coefficient in front of \(x\), which in this case is \(-2\).
• Divide this coefficient by 2, which yields \(-1\).
• Square the result from the previous step, which results in 1.
• Add and subtract this squared number inside the equation to complete the square.

This process is applied to both \(x\) and \(y\) terms separately, transforming the equation into the form \((x-a)^2\) or \((y-b)^2\), making it easier to identify the center and radius of the circle.

Center of a circle

The center of a circle is a key component in understanding its position on a graph. Represented by the coordinates \((h, k)\) in the standard circular equation \((x - h)^2 + (y - k)^2 = r^2\), the center is the point equidistant from all points on the circle's circumference.
When identifying the center from an equation that initially doesn’t appear in the standard form, such as having initial forms like: \(x^2 + y^2 - 2x - 2y = 2\), steps like completing the square are used. This converts the equation to make it clear what \(h\) and \(k\) are. In our example, after rearranging and completing the square, the equation is transformed to \((x - 1)^2 + (y - 1)^2 = 4\).
Here, the circle's center is evident as \((1, 1)\). Thus, the center is not only a theoretical component but a practical tool for graphically plotting the circle accurately.

Radius of a circle

The radius of a circle is the linear measurement from its center to any point on its outer edge. In the standard form equation \((x - h)^2 + (y - k)^2 = r^2\), the term \(r^2\) represents the square of the radius.

• To find the radius \(r\), one would simply take the square root of \(r^2\).
• The equation can often have \(r^2\) in an expanded form that needs simplification after completing the square.

For example, if completing and simplifying an equation leads us to \((x - 1)^2 + (y - 1)^2 = 4\), it shows that \(r^2 = 4\). Hence, the circle's radius \(r\) is \(\sqrt{4} = 2\).
Understanding the radius is crucial for drawing solutions and problems involving circles, influencing area calculations and other geometric properties.