Problem 61 - College Algebra

Question

Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$ x^{2}+y^{2}-x+2 y+1=0 $$

Step-by-Step Solution

Answer

The equation in standard form is \((x - 0.5)^2 + (y+1)^2 = 0.25\). The center of the circle is at (0.5, -1) and the radius is 0.5.

Step 1: Group Like Terms

Group the like terms together. The given equation is \(x^{2}+y^{2}-x+2y+1=0\). Reorder the equation: \(x^2 - x + y^2 + 2y + 1 = 0\)

Step 2: Complete the Square for x

Complete the square for the x terms: \(x^2 - x\). Divide the coefficient of x by 2 and square the result: \((-1/2)^2 = 0.25\). Add and subtract this from the equation: \[(x^2 - x + 0.25) - 0.25 + y^2 + 2y + 1 = 0\]. Now, the x-term can be expressed as a perfect square: \([(x - 0.5)^2 - 0.25 + y^2 + 2y + 1 = 0\]

Step 3: Complete the Square for y

Similar to step 2, complete the square for the y term. Divide the coefficient of y by 2 and square the result, \(1^2 = 1\). Add and subtract this from the equation: \([(x - 0.5)^2-0.25 + (y^2 + 2y +1) -1 + 1 = 0\]. Now, the y-term can be expressed as a perfect square: \[(x - 0.5)^2-0.25 + (y+1)^2 -1 + 1 = 0\]

Step 4: Simplify

Combine the constant terms and reorder the equation: \[(x - 0.5)^2 + (y+1)^2 -0.25 = 0\]. Therefore, \[(x - 0.5)^2 + (y+1)^2 = 0.25\]

Step 5: Identify Center and Radius

Now the equation looks like \((x-a)^2 + (y-b)^2 = r^2\). So, the center of the circle is at (0.5, -1) and the radius is the square of 0.25 = 0.5.

Step 6: Graph the Equation

Using the center (0.5, -1) and radius 0.5, draw a circle on a graphical representation. The center is at (0.5, -1) and the radius is 0.5

Key Concepts

Equation of a Circle

In algebra, the equation of a circle offers a beautiful and symmetrical expression of a fundamental geometric shape. A circle in the Cartesian coordinate plane can typically be expressed in a special form, which highlights its characteristics succinctly.
In general, any circle can be described by the equation \[(x - a)^2 + (y - b)^2 = r^2\]where:

• \((x, y)\) represent the coordinates of any point on the circle.
• \((a, b)\) is the center of the circle, which is equidistant from all points on the circle.
• \(r\) is the radius, indicating how far points on the circle are from the center.

Completing the square is a useful algebraic technique to convert an equation of a circle from its expanded form into this standard form.

Standard Form

The standard form of a circle's equation highlights its symmetry and simpleness. For any circle, the standard form equation is \[(x - a)^2 + (y - b)^2 = r^2\].This equation clearly shows both the center and the radius of the circle, helping significantly in graphing and understanding the circle's properties.
The benefits of the standard form are:

• Easy identification of the circle's center \((a, b)\).
• Direct calculation of the circle's radius \(r\), simplifying graphing.
• Straightforward transformation from other forms of the equation using algebraic techniques like completing the square.

Always make sure to simplify any equation into this form to gain insights into the circle's geometric properties.

Center and Radius

Knowing the center and radius of a circle is crucial for understanding its position and size. The center of a circle is represented as a point \((a, b)\), which is the fixed midpoint that every point on the circle's edge is equidistant from.
The radius is the constant distance from the center to any point on the circle. This distance defines the size of the circle. Perfect identification of these parameters allows you to sketch the circle accurately on the plane.
To extract the center and radius from an algebraic expression:

• Reorganize the equation into standard form.
• Identify \((a, b)\) from \((x-a)^2 + (y-b)^2\) as the center.
• Compute the radius \(r\) as the square root of the constant term, ensuring positivity.

For example, from the equation \((x - 0.5)^2 + (y + 1)^2 = 0.25\),the center is \((0.5, -1)\) and the radius is 0.5.

Graphing Circles

Graphing circles requires only the center and radius to draw an accurate representation.
Once you have the center \((a, b)\) and radius \(r\), you can visualize and sketch the circle easily.

• Start by plotting the center point on the coordinate plane.
• Use a compass or an estimation method to mark all positions that are exactly \(r\) units away from the center.
• Create a smooth, continuous curve through these points to form your circle.

Tools like graph paper or digital graphing calculators can assist greatly in accurate graphing.
Always double-check the radius for a perfect curve, and confirm symmetry relative to the center. That ensures the most precise graphing possible.