Graph: Center and radius Equation: Center and radius
Practice makes perfect
In this exercise we will determine the least amount of information to graph and to write an equation of a circle. Let's first recall the definition of a circle.
A circle is a set of all points on a plane that are equidistant from a given point.
The given point is the center of the circle. The distance between the center and any point on the graph is the radius of the circle. Now, we will begin by considering how to graph a circle.
Graph of a Circle
Let's consider the construction of a circle. Remember that a circle can be graphed by using a compass. We first locate the center on the coordinate plane.
Next, we put the compass point on the center and widen the arms of the compass to the given radius.
Then, we can draw the circle!
To draw a circle, four sets of information can be provided.
However, by the definition and construction of a circle, we should first identify the center of the circle and then we should determine its radius. Therefore, we can conclude that the least amount of information to graph a circle is the center and the radius.
Equation of a Circle
The equation of a circle can be written by having provided three sets of information.
Center and radius of a circle
Center of and a point on a circle
Diameter of a circle with its endpoints
To determine the least amount of information let's recall Theorem 12-16, the standard form of an equation of a circle.
Theorem 12-16
An equation of a circle with center (h,k) and radius r is (x-h)^2+(y-k)^2=r^2.
With this theorem, we can conclude that even if we have been given the second and third sets of information, we should first determine the center and radius of the circle. Thus, the least amount of information to write the equation of a circle is also the center and radius.
