91Ó°ÊÓ

Completing the square is a method used to simplify quadratic equations, making them easier to work with. It's often used to convert quadratic equations into a standard form that is easier to analyze and solve. To complete the square, follow these steps:

1. **Group and factor:** Start by grouping the terms involving the same variable and factor out any coefficients of the quadratic term. For example, in the equation given, we grouped the x and y terms and factored out the 9 from each group:
\[ 9(x^2 - \frac{2}{3}x) + 9(y^2 + \frac{2}{3}y) - 23 = 0 \]

2. **Find the perfect square trinomial:** For each group, add and subtract the appropriate constant to form a perfect square trinomial. For the x terms, this involves finding (half of -2/3)^2 and for the y terms, finding (half of 2/3)^2:
\[ 9[(x^2 - \frac{2}{3}x + (\frac{-2}{6})^2 - (\frac{-2}{6})^2] + 9[(y^2 + \frac{2}{3}y + (\frac{2}{6})^2 - (\frac{2}{6})^2] = 23 \]

3. **Rewrite as squared binomials:** Convert each group into squared binomials while keeping the balance of the equation:
\[ 9[(x - \frac{1}{3})^2 - \frac{1}{9}] + 9[(y + \frac{1}{3})^2 - \frac{1}{9}] - 23 = 0 \]

Conic sections are curves obtained by slicing a cone with a plane. They include circles, ellipses, parabolas, and hyperbolas. Each type of curve results from a different angle and position of the plane. The circle is a special conic section where the plane cuts the cone parallel to its base.

**Properties of circles:**

• All points on a circle are equidistant from its center.
• The distance from the center to any point on the circle is called the radius.
• The diameter is twice the radius.

**Identifying Circles within Equations**
To identify if an equation represents a circle, the general form of the circle’s equation \[ (x - h)^2 + (y - k)^2 = r^2 \]

By comparing this form to the equation after completing the square, it becomes easier to identify the circle's center and radius. For instance, in the exercise, our simplified equation:
\[ (x - \frac{1}{3})^2 + (y + \frac{1}{3})^2 = \frac{25}{9} \]
shows a circle with center \[ (\frac{1}{3}, -\frac{1}{3}),\] and radius \[ \frac{5}{3}\].

The standard circle equation is a special form of the conic section equations. It's written as:
\[ (x - h)^2 + (y - k)^2 = r^2 \]

where:

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

**Converting to Standard Form**
To convert an equation to the standard form, follow these steps:

• Isolate the x and y terms and move the constant term to the other side of the equation.
• Complete the square for both x and y groups as shown in the section about completing the square.
• Rewrite the equation in the standard form.
• Identify the values of h, k, and r, and you have the circle's center and radius.

Using the example from the exercise, we converted
\[ 9(x - \frac{1}{3})^2 + 9(y + \frac{1}{3})^2 - 25 = 0 \]
into the standard form:
\[ (x - \frac{1}{3})^2 + (y + \frac{1}{3})^2 = \frac{25}{9} \]
which clearly shows the center at \((\frac{1}{3}, -\frac{1}{3})\), and the radius as \(\frac{5}{3}\).