Problem 24 - Algebra 2 and Trigonometry

Question

In \(20-27\) : a. Write each equation in center-radius form. b. Find the coordinates of the center. . Find the radius of the circle. $$ x^{2}+y^{2}+6 x-6 y+6=0 $$

Step-by-Step Solution

Answer

Center: (-3, 3); Radius: \(2\sqrt{3}\).

Step 1: Rearrange the Equation

Start with the given equation: \[ x^2 + y^2 + 6x - 6y + 6 = 0 \]We need to rearrange terms by grouping the x terms and y terms together:\[ x^2 + 6x + y^2 - 6y + 6 = 0 \]

Step 2: Complete the Square for x

To complete the square for the terms involving x:- Take the coefficient of x, which is 6, divide by 2, and square it: \[ \left( \frac{6}{2} \right)^2 = 3^2 = 9 \]- Add and subtract 9 inside the equation:\[ x^2 + 6x + 9 + y^2 - 6y + 6 = 0 + 9 \]This simplifies the expression to:\[ (x + 3)^2 \]

Step 3: Complete the Square for y

Now, complete the square for the terms involving y:- Take the coefficient of y, which is -6, divide by 2, and square it:\[ \left( \frac{-6}{2} \right)^2 = (-3)^2 = 9 \]- Add and subtract 9 inside the equation:\[ x^2 + 6x + 9 + y^2 - 6y + 9 = 0 + 9 + 9 \]This simplifies the expression to:\[ (y - 3)^2 \]

Step 4: Simplify and Write the Equation in Center-Radius Form

We now have:\[ (x + 3)^2 + (y - 3)^2 = 12 \]This is the center-radius form of the circle equation.

Step 5: Identify the Center of the Circle

From the center-radius form \((x + 3)^2 + (y - 3)^2 = 12\), we identify the center:The center is at \(-3, 3\).

Step 6: Calculate the Radius of the Circle

The equation \((x + 3)^2 + (y - 3)^2 = 12\) shows that the radius squared is 12.So, the radius is:\[ \text{Radius} = \sqrt{12} = 2\sqrt{3} \]

Key Concepts

Completing the Square

Completing the square is a technique used to transform quadratic equations into a form that makes certain values, such as the position of vertices in parabolas and centers in circles, easy to identify. The process focuses on manipulating an equation into a perfect square trinomial. For example, consider the expression involving x in our equation:

• The expression is initially \(x^2 + 6x\).
• To complete the square, take the coefficient of x (which is 6), divide it by 2, resulting in 3, and then square it, giving 9.
• Add and subtract this value (9) in the equation to keep it balanced.

So, \(x^2 + 6x\) becomes \((x + 3)^2 - 9\). This technique is similarly applied to the y-term, converting \(y^2 - 6y\) to \((y - 3)^2 - 9\). Completing the square helps in transforming the equation into a format suitable for identifying characteristics of conic sections like circles.

Circle Equation

The equation of a circle in center-radius form is typically expressed as:\[(x - h)^2 + (y - k)^2 = r^2\]Here, \((h, k)\) represents the coordinates of the circle's center, and \(r\) is its radius. This form makes it easy to visualize the circle's positioning and size. For the given equation, we started with the original expression:

• Through completing the square for both x and y terms and rearranging, it becomes \((x + 3)^2 + (y - 3)^2 = 12\).

This is the center-radius form of the equation, where the expressions \((x + 3)\) and \((y - 3)\) allow us to easily find the center of the circle.

Coordinates of the Center

In the center-radius form of a circle equation, the coordinates of the center are crucial for understanding the circle's position on a coordinate plane. The equation for our problem is:\[(x + 3)^2 + (y - 3)^2 = 12\].In this form:

• The standard pattern is \((x - h)^2 + (y - k)^2 = r^2\).
• Rewriting \((x + 3)^2\) as \((x - (-3))^2\) shows that the x-coordinate of the center is \(-3\).
• Similarly, \((y - 3)^2\) fits the pattern directly, showing the y-coordinate of the center is 3.

Thus, the circle's center is at the point \((-3, 3)\). Knowing the center is essential for sketching or analyzing the circle in geometric tasks.

Radius of the Circle

Determining the radius of the circle involves interpreting the simplified center-radius form equation. The completed equation is:\[(x + 3)^2 + (y - 3)^2 = 12\].Here, the right side of the equation represents \(r^2\) where \(r\) is the radius:

• Since \(r^2 = 12\), we find \(r\) by taking the square root of 12.
• This gives \(r = \sqrt{12}\).
• We can simplify this further to \(r = 2\sqrt{3}\).

The radius is a measure of how far any point on the circumference is from the center and is vital for understanding the circle's size. In this case, the radius tells us about the circle's span from the center point \((-3, 3)\).