A circle is a geometrical shape. It is defined as having a center, and being the set of all points that are a certain fixed distance from that center. (The fixed distance is called the radius of the circle.) The circle is not of much use in algebra since the equation of a circle isn't a function.
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In primative terms (that is, in physical and non-algebraic terms), a circle was a geometric shape formed in sand with two sticks and some string.
Once you've pulled the second stick all the way around once, the trace of the second stick is the circle.
In algebraic terms, a circle is the figure formed by the set (or "locus") of points (x, y) at some fixed distance r from some fixed point (h, k). The value of r is called the "radius" of the circle, and the point (h, k) is called the "center" of the circle.
The general equation of a circle is:
x2 + y2 + Dx + Ey + F = 0
The "center-radius" form of the equation is:
(x − h)2 + (y − k)2 = r2
...where the h and the k come from the center point (h, k) and the r2 comes from the radius value r.
If the center is at the origin, so (h, k) = (0, 0), then the equation simplifies to x2 + y2 = r2.
Note: The general form may be given in your book using different letters for the coefficients, but the center-radius form will use the same letters as shown above. The center is always denoted by (h, k) and the radius is always denoted by r.
Your textbook might refer to something called the "standard" form of the circle equaion. Unfortunately, "standard form" has no standard meaning that I've been able to determine, so you'll have to keep track of what your particular textbook intends by that term.
The center-radius form of the circle equation comes directly from the Distance Formula and the definition of a circle. If the center of a circle is the point (h, k) and the radius is length r, then every point (x, y) on the circle is distance r from the point (h, k). Plugging this information into the Distance Formula (using r for the distance between the points and the center), we get:
You should practice until you can readily recognize this formula, because you will be expected to be able to read information from it.
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To draw (that is, to graph) a circle in algebra (as opposed to drawing one in geometry, where you can use a compass), follow these steps:
Here's an example of what that looks like:
The y2 term in the equation means the same thing as (y − 0)2, so the equation is really the following:
(x − 2)2 + (y − 0)2 = 52
Since the circle's equation is now in center-radius form, I can read the values directly: The center must be at (h, k) = (2, 0), and the radius must be r = 5.
To sketch, I'll first draw the dot for the center:
I can't draw a perfect circle, so I'll work in steps. First, in pencil, I'll rough in markers that are five units away from the center in each of the four easy directions (that is, directly up, down, left, and right from the center):
Then I'll rough in the curvy bits in between these markers, turning the paper as I go:
I'll make whatever corrections look useful, trying to make my circle look properly circular, and ink my final answer as a solid dark line:
So that I'm handing in a nice-looking drawing, I'll erase the pencil marks from my roughing-in, and I'll copy my center and radius values that I found earlier.
center: (h, k) = (2, 0)
radius: r = 5
Hint: It can be very helpful to rotate your paper when you do your drawing, so as to end up with a circle that looks fairly round.
You can use the Mathway widget below to practice finding the center and radius of a circle (or skip the widget and continue on to the next page). Try the entered exercise, or type in your own exercise. Then click the button to compare your answer to Mathway's.
(Click "Tap to view steps" to be taken directly to the Mathway site for a paid upgrade.)