Problem 102 - College Algebra and Calculus: An Applied Approach

Question

An equation of a circle is written in standard form. Indicate the coordinates of the center of the circle and determine the radius of the circle. Rewrite the equation of the circle in general form. \(\left(x-\frac{1}{2}\right)^{2}+(y-2)^{2}=7\)

Step-by-Step Solution

Answer

The coordinates of the center of the circle are \(\left(\frac{1}{2}, 2\right)\) and the radius is approximately 2.646. The equation of the circle in general form is \(x^2+y^2-x-4y+3 = 0\).

Step 1: Identify the Center

From the given equation, \((x-\frac{1}{2})^2 + (y-2)^2 = 7\), the center of the circle can be determined by identifying the values of \(h\) and \(k\) in the form \((x-h)^2 + (y-k)^2 = r^2\). This gives us a center at \(\left(\frac{1}{2}, 2\right)\).

Step 2: Determine the Radius

The radius \(r\) of the circle can be determined from the same standard form. Here, the number on the right-hand side of the equation (7) represents \(r^2\), the square of the radius. Taking the square root of this number gives us the radius, which is approximately \(\sqrt{7}\) or approximately 2.646.

Step 3: Rewrite in General Form

The general form of the equation of a circle is \(x^2+y^2+2gx+2fy+c=0\). Here, \(g = -h\), \(f = -k\), and \(c = h^2 + k^2 - r^2\). Substituting the values from our equation gives us \(x^2+y^2-1x-4y+3 = 0\)

Key Concepts

Standard Form

The standard form of a circle's equation is a powerful tool in geometry and algebra that makes it easy to identify the circle's key characteristics. This form of the equation is typically written as:\[(x-h)^2 + (y-k)^2 = r^2\]where:

• \( (h, k) \) are the coordinates of the center of the circle.
• \( r \) is the radius of the circle.

By looking at a circle in this form, you can directly extract the center and radius without needing to rearrange or perform additional calculations. This form is particularly useful for visualizing and graphing circles, as it's designed to match the formula for the distance squared in the Euclidean plane. Using this structure, you can easily compare different circles and understand their relative positions and sizes.

Center of a Circle

The center of a circle is a critical point that helps define the circle’s location. In the equation of a circle in standard form, the center is given by the point \((h, k)\). The coordinates \(h\) and \(k\) are found directly from the transformation applied to \(x\) and \(y\).
For example, in the standard circle equation:\[(x - \frac{1}{2})^2 + (y - 2)^2 = 7\]the terms \(x - \frac{1}{2}\) and \(y - 2\) indicate that the center of the circle is at \((h, k) = (\frac{1}{2}, 2)\).
It is important to note that the signs of \(h\) and \(k\) are opposite to what you see in the equation. This is due to the subtraction in \((x-h)\) and \((y-k)\), meaning if the equation shows \(x - h\), the center's \(x\)-coordinate is actually \(h\). Understanding this will help in graphing the circle correctly on a coordinate plane.

Radius of a Circle

Finding the radius in the standard form equation is quite straightforward. The value on the right-hand side of the equation represents \(r^2\), where \(r\) is the radius of the circle. To find the actual radius, take the square root of this number.
For instance, in the equation:\[(x - \frac{1}{2})^2 + (y - 2)^2 = 7\]the value 7 is \(r^2\). By solving:\[r = \sqrt{7}\]This calculation tells us that the radius is approximately 2.646 units long. Knowing the radius helps not only in drawing the circle but also in understanding the circle's scale relative to other figures in a given problem. Its precision is crucial especially when dealing with geometric constructions and measurements.

General Form of a Circle

The general form of the equation of a circle differs from the standard form but is essential in various algebraic operations. This form is expressed as:\[x^2 + y^2 + 2gx + 2fy + c = 0\]where the coefficients \(g\), \(f\), and constant \(c\) are derived from the circle's standard form parameters.

• In practice, \(g = -h\) and \(f = -k\).
• The constant \(c\) can be calculated as: \[c = h^2 + k^2 - r^2\]

To convert from standard to general form, one must expand the squared terms and then collect like terms. In the example, starting from\[(x - \frac{1}{2})^2 + (y - 2)^2 = 7\]transforms to: \[x^2 + y^2 - x - 4y + 3 = 0\].This general form, although less intuitive for graphing, is useful in proving geometric properties and solving intersection problems.