Question

Write the standard form of the equation and the general form of the equation of each circle of radius \(r\) and center \((h, k)\). Graph each circle. $$ r=\frac{1}{2} ;(h, k)=\left(0,-\frac{1}{2}\right) $$

Short answer

The standard form is \[ x^2 + \left(y + \frac{1}{2}\right)^2 = \frac{1}{4} \] and the general form is \[ x^2 + y^2 + y = 0. \]

Step-by-step solution

Identify the standard form of a circle's equation

The standard form of the equation of a circle is \( (x - h)^2 + (y - k)^2 = r^2 \). Here, \( (h, k) \) is the center, and \( r \) is the radius.

Substitute the given values into the standard form

Given \( r = \frac{1}{2} \) and \( (h, k) = (0, -\frac{1}{2}) \), substitute these into the standard form equation: \[ (x - 0)^2 + \left(y + \frac{1}{2}\right)^2 = \left(\frac{1}{2}\right)^2. \] This simplifies to \[ x^2 + \left(y + \frac{1}{2}\right)^2 = \frac{1}{4}. \]

Expand to find the general form

To convert from the standard form to the general form, expand \( (y + \frac{1}{2})^2 \): \[ y^2 + y + \frac{1}{4}. \] Therefore, the equation becomes \[ x^2 + y^2 + y + \frac{1}{4} = \frac{1}{4}. \]

Isolate all terms to one side

To express the equation in general form, move all terms to one side to set the equation equal to zero: \[ x^2 + y^2 + y + \frac{1}{4} - \frac{1}{4} = 0. \] This simplifies to \[ x^2 + y^2 + y = 0. \]

Graph the circle

To graph the circle, plot the center at \( (0, -\frac{1}{2}) \), and use the radius \( \frac{1}{2} \) to draw the circle around the center.

Key concepts

Standard Form

The standard form of a circle's equation is a critical concept in geometry. It allows us to understand and graph circles in a clear and straightforward way. The standard form is given by the equation:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Here,

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

Let's break this down with the given values \(r=\frac{1}{2}\) and \((h, k)=\big(0,-\frac{1}{2}\big)\). Substituting these into the standard form equation, we get:
\[ (x - 0)^2 + \big(y + \frac{1}{2}\big)^2 = \big(\frac{1}{2}\big)^2 \]
Simplifying further, the equation of the circle becomes: \[ x^2 + \big(y + \frac{1}{2}\big)^2 = \frac{1}{4} \]
This equation tells us that the circle has a center at \( (0, -\frac{1}{2}) \) and a radius of \( \frac{1}{2} \).

General Form

The general form of a circle's equation provides us with another way to represent the same circle in a more expanded format. To convert from the standard form to the general form, follow these steps.

Taking our standard form equation:
\[ x^2 + \big(y + \frac{1}{2}\big)^2 = \frac{1}{4} \], we need to expand and rearrange it.
First, expand \( \big(y + \frac{1}{2}\big)^2 \):
\[ \big(y + \frac{1}{2}\big)^2 = y^2 + y + \frac{1}{4} \]
Substituting back in, the equation becomes:
\[ x^2 + y^2 + y + \frac{1}{4} = \frac{1}{4} \]
To write this in general form, move all terms to one side of the equation:
\[ x^2 + y^2 + y + \frac{1}{4} - \frac{1}{4} = 0 \]
This simplifies to:
\[ x^2 + y^2 + y = 0 \]
The general form of the equation of our circle is \( x^2 + y^2 + y = 0 \). By understanding how to manipulate the equations, we can see that both forms represent the same circle.

Radius and Center

Understanding the radius and center is key to solving circle equations. These two components define the size and location of the circle on the coordinate plane.

• The center of the circle is given by the coordinates \( (h, k) \).
• The radius is the distance from the center to any point on the circle, denoted by \( r \).

For our example, the center is \((0, -\frac{1}{2})\) and the radius is \(\frac{1}{2}\). These values are essential when substituting into the standard form equation.
When graphed, the center is a fixed point on the coordinate plane, with the circle extending outwards uniformly in all directions for the radius distance. To visualize this, plot the point \((0, -\frac{1}{2})\) as the center, then measure a distance of \(\frac{1}{2}\) units in all directions around this point, and draw the circle.
Recognizing how the radius and center come together in the equation helps in accurately graphing and understanding the circle's properties.