91Ó°ÊÓ

Completing the square is a method used to make a quadratic expression into a perfect square trinomial. This technique is particularly useful in geometry for converting a circle equation into its standard form.
To complete the square for a term like \(x^{2} + bx\), follow these steps:

• Take the coefficient of x, which is b. Then divide it by 2.
• Square the result to find the number that you will add and subtract. This ensures that we do not change the value of the original equation.

For example, in the expression \(x^2 + 8x\), divide 8 by 2 to get 4. Squaring 4 gives 16, so we add and subtract 16 within the equation.
Thus, \(x^2 + 8x\) becomes \(x^2 + 8x + 16 - 16 = (x + 4)^2 - 16\). This process is then repeated in a similar manner for the y terms.

The standard form of a circle's equation reveals important characteristics of the circle, such as its center and radius. The standard form is written as:
\[(x-h)^2 + (y-k)^2 = r^2\]
where (h, k) is the center of the circle and r is its radius.
To convert a general circle equation to standard form, completing the square is often required for both x and y terms. This reveals the hidden structure of the equation. Let's take the example:
\[x^2 + 8x + y^2 - 10y = 0\]
We complete the square for both x and y terms to transform it:
\[(x + 4)^2 - 16 + (y - 5)^2 - 25 = 0\]
By reorganizing and simplifying:
\[(x + 4)^2 + (y - 5)^2 = 41\]
We now have the equation in standard form, making it easier to identify the circle's characteristics.

From the standard form of the circle equation,\[(x-h)^2 + (y-k)^2 = r^2\], we can directly read off the center (h, k) and the radius r.

Consider the equation derived previously: \[(x + 4)^2 + (y - 5)^2 = 41\]. The equation suggests:

• The center (h, k) is \(-4, 5\).
• The radius r is the square root of 41.

To identify these, compare the standard form to the given equation:

• The term \((x+4)\) indicates a shift left by 4 units.
• The term \((y-5)\) indicates a shift up by 5 units.

Thus, the center becomes (-4, 5).
If the right-hand side is 41, then the radius is found by taking the square root of 41: \(\r = √41\). This method ensures we have a complete understanding of the circle's properties through its equation.