Problem 80 - A Graphical Approach to Precalculus with Limits

Question

Find the center-radius form of the circle with the given equation. Determine the coordinates of the center, find the radius, and graph the circle. $$x^{2}+y^{2}-10 x+8 y+5=0$$

Step-by-Step Solution

Answer

Center: (5, -4); Radius: 6.

Step 1: Identify the Equation Form

The given equation is in the general form of a circle, which is: \[ x^2 + y^2 + Dx + Ey + F = 0 \]Here, \(D = -10\), \(E = 8\), and \(F = 5\). Our task is to convert this into the center-radius form.

Step 2: Complete the Square for x

To convert the equation, start by completing the square for the \(x\) terms. Take the coefficient of \(x\), which is \(-10\), divide by 2, and square it:\[ \left( \frac{-10}{2} \right)^2 = 25 \]Add and subtract 25 within the equation to complete the square for \(x\).

Step 3: Complete the Square for y

Next, complete the square for the \(y\) terms. Take the coefficient of \(y\), which is \(8\), divide by 2, and square it:\[ \left( \frac{8}{2} \right)^2 = 16 \]Add and subtract 16 within the equation to complete the square for \(y\).

Step 4: Rearrange to Center-Radius Form

Incorporate the completed squares into the equation:\[ (x^2 - 10x + 25) + (y^2 + 8y + 16) = 25 + 16 - 5 \]\[ (x - 5)^2 + (y + 4)^2 = 36 \]The equation is now in center-radius form: \[ (x - h)^2 + (y - k)^2 = r^2 \] where \(h = 5\), \(k = -4\), and \(r^2 = 36\).

Step 5: Identify Center and Radius

From the center-radius form \((x - 5)^2 + (y + 4)^2 = 36\), the center of the circle is \((h, k) = (5, -4)\), and the radius is \(r = \sqrt{36} = 6\).

Step 6: Graph the Circle

To graph the circle, plot the center at \((5, -4)\).Use the radius of 6 to mark points 6 units away from the center in all directions to form the circle.

Key Concepts

Completing the Square

Completing the square is a useful technique in algebra that helps transform quadratic equations into a form that reveals essential features. When dealing with circle equations, you often start with a general form \[ x^2 + y^2 + Dx + Ey + F = 0 \]. To convert it to a form that describes a circle, you need to "complete the square" for both the \(x\) and \(y\) terms.

• For the \(x\)-terms, take the coefficient of \(x\) (often noted as \(D\)), divide it by 2, and then square the result.
• Add and subtract this squared value within the equation.

Repeat these steps for the \(y\)-terms using its coefficient (\(E\)). This process effectively turns mixed terms into square terms, allowing you to rearrange the equation into a more easily identifiable form. This transformation is crucial for understanding the geometric properties of the circle they represent, as it clearly shows the center and radius after further rearrangement.

Center-Radius Form

The center-radius form of a circle's equation is \[ (x - h)^2 + (y - k)^2 = r^2 \], where \((h, k)\) represents the center and \(r\) is the radius. This form is incredibly helpful because it gives a clear visual understanding of where the circle is located on a coordinate plane and how large it is.

• Once you complete the square for both variables, the equation can be written in this form. This is accomplished by grouping the \(x\) and \(y\) terms into perfect squares.
• The coordinates \((h, k)\) are directly derived from the expressions in each squared term. For example, if you have \((x - 5)^2\), the \(x\)-coordinate of the center is 5.

In the center-radius form, the constant on the right side of the equation is the square of the radius. If this equals a number like 36, then the radius, \(r\), would be the square root of that number, giving \(r = 6\). This form simplifies understanding key aspects of the circle and helps in graphing.

Graphing Circles

Graphing circles becomes straightforward once you have transformed the circle's equation into the center-radius form. This method provides everything needed to sketch the circle accurately on a coordinate grid.

• Begin by plotting the center of the circle; using the equation, locate \((h, k)\) on your graph. These are your central reference points.
• Next, use the radius \(r\) to measure the distance from the center in all directions. You can mark several key points, such as to the right, left, above, and below the center, each at a distance of \(r\) from the center.

Connect these points smoothly to sketch the circular shape. Graphing in this way ensures an even and accurate representation of the circle on the grid. This clear visual can aid in understanding the circle's position and size relative to other elements on the plane.