91Ó°ÊÓ

Completing the square is a technique used to simplify quadratic expressions, making them easier to work with. It's especially useful in rewriting circle equations in a form that reveals the circle's center and radius. When we deal with an equation like \(x^2 + y^2 - \frac{x}{2} - \frac{3y}{2} + \frac{3}{8} = 0\), we want to rewrite it by completing the square for both \(x\) and \(y\) terms.

Completing the square involves a few steps. First, separate the terms: for \(x\), take \(x^2 - \frac{x}{2}\) and for \(y\), take \(y^2 - \frac{3y}{2}\). We focus on making these perfect square trinomials.

• For the \(x\)-terms, we take half of \(-\frac{1}{2}\), which is \(-\frac{1}{4}\), and square it to get \(\frac{1}{16}\).
• For the \(y\)-terms, take half of \(-\frac{3}{2}\), which is \(-\frac{3}{4}\), and square it to get \(\frac{9}{16}\).

Adding and subtracting these squares helps us form expressions like \((x - \frac{1}{4})^2\) and \((y - \frac{3}{4})^2\). This transformation represents a shift to a standard form.

The standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\). Here, \((h, k)\) represents the center of the circle, and \(r\) is the radius. By converting our equation \(x^2 + y^2 - \frac{x}{2} - \frac{3y}{2} = -\frac{3}{8}\) after completing the square, we aim to express it in this format.

After completing the square, we rewrite the equation so that it looks like this: \((x - \frac{1}{4})^2 + (y - \frac{3}{4})^2 = 1\).

• This form clearly parallels the standard form, and comparing them gives us direct information about the circle.
• The terms \((x - \frac{1}{4})^2\) and \((y - \frac{3}{4})^2\) indicate we have succeeded in forming perfect squares.

The completion process organized the terms and adjusted the constant on the right-hand side to complete the transformation into the standard form.

Identifying the center and radius of a circle from its equation is straightforward once the equation is in standard form \((x - h)^2 + (y - k)^2 = r^2\). Here, \((h, k)\) is the center and \(r\) is the radius.

From the transformed equation \((x - \frac{1}{4})^2 + (y - \frac{3}{4})^2 = 1\), you can easily see:

• The center \((h, k)\) is \((\frac{1}{4}, \frac{3}{4})\).
• The circle's radius \(r\) is \(\sqrt{1} = 1\).

This identification helps in visualizing the circle's position and size on a coordinate plane. The radius signifies how far any point on the circle is from the center, highlighting its spherical symmetry.