Problem 61 - Precalculus Essentials

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 standard form of the equation is \((x-0.5)^2 + (y+1)^2 = 1.25\). The center of the circle is at (0.5, -1), and the radius is \(sqrt{1.25}\)

Step 1: Rearrange Terms

First, rearrange terms to group \(x\) and \(y\) terms separately. The equation will become: \(x^{2}-x+y^{2}+2y+1=0\)

Step 2: Complete the Square - x terms

To complete the square, half the coefficient of the 'x' term, square it and add to both sides. Half of -1 is -0.5, providing \(0.5^2 = 0.25\). The equation will become: \(x^{2}-x+0.25+y^{2}+2y+1=0+0.25\). We simplify this to: \((x-0.5)^2+y^{2}+2y+1=0.25\)

Step 3: Complete the Square - y terms

To complete the square for the 'y' terms, half of 2 is 1 so we add \(1^2 = 1\) to both sides. The equation will become: \((x-0.5)^2+(y^{2}+2y+1)+1=0.25+1\), which will further simplify to: \((x-0.5)^2 + (y+1)^2 = 1.25\)

Step 4: Find Center and Radius

After simplifying, the standard form is obtained as \( (x- h)^2 + (y - k)^2 = r^2 \), where \(h = 0.5\), \(k = -1\), and \(r = sqrt{1.25}\). Hence, center of the circle is at (0.5, -1) and radius is \(sqrt{1.25}\)

Key Concepts

Standard Form of a Circle

When dealing with circles in coordinate geometry, the standard form equation is a reliable starting point to visualize and solve problems. The standard form for the equation of a circle is given by \( (x - h)^2 + (y - k)^2 = r^2 \) where \(h\) and \(k\) represent the coordinates of the center of the circle, and \(r\) is its radius.

To achieve this standard form when given an expanded equation, as in our exercise \(x^2 + y^2 - x + 2y + 1 = 0\), the process called completing the square is employed. This involves rearranging and adjusting terms to create perfect square trinomials for the \(x\) and \(y\) components separately. Adding the same values to each side of the equation to maintain equality ensures that the circle's equation is transformed without altering its actual parameters. We then clearly see the values for \(h\), \(k\), and \(r\) once the equation is in standard form.

Center of a Circle

The center point of a circle is effectively the 'anchor' around which the circle is defined. In our standard form equation, \( (x - h)^2 + (y - k)^2 = r^2 \) the center is given by the coordinates \( (h, k) \).

• Why is it important? The center is crucial in geometry as all points on the circle are equidistant from this central point. That's the essence of circular symmetry!
• Impact on graphing: Knowing the center is the first step in plotting a circle on a coordinate plane. Once you mark the center, you can use the radius to draw the circle accurately.

Through completing the square, as shown in the solution, we isolate the center's coordinates from an equation that might initially seem complex. This simplification is key for understanding and accurately depicting circles mathematically.

Radius of a Circle

The radius of a circle is arguably its most fundamental characteristic. It's the straight-line distance from the center of the circle to any point on its circumference. In the standard form \( (x - h)^2 + (y - k)^2 = r^2 \), the radius \(r\) can be determined once the equation is properly arranged.

The radius dictaminates the size of the circle and is vital for calculations involving arc length, sector area, and more. Upon completing the square in the given exercise, we identified \( sqrt{1.25} \) as the radius. This value allows us to not only understand the size of the circle but to perform a variety of subsequent tasks, including finding the circumference \( (2\pi r) \) and the area \( (\pi r^2) \) of the circle.

Precalculus Graphing

Graphing is a foundational skill in precalculus that brings equations to life. When graphing a circle, understanding how to apply the standard form equation to plot its shape on a coordinate plane is crucial.

• Basics: Plot the center of the circle first, then use the radius to determine how far from the center the circle extends in all directions.
• Points on Circumference: With the radius, you can plot points that lie on the circle's circumference at regular intervals around the center and connect them to form the shape.

Graphing does more than just portray geometrical figures; it helps in visualizing and solving more complex problems involving circles, such as finding intersections with other shapes or lines. As shown in the exercise, graphing reinforces the relationship between an algebraic equation and its geometric representation.