Conics: Circles: Introduction & Drawing (page 1 of 3)
A circle is a geometrical shape, and is not of much use in algebra, since the equation of a circle isn't a function. But you may need to work with circle equations in your algebra classes.
In "primative" terms, a circle is the shape formed in the sand by driving a stick (the "center") into the sand, putting a loop of string around the center, pulling that loop taut with another stick, and dragging that second stick through the sand at the further extent of the loop of string. The resulting figure drawn in the sand is a circle.
In algebraic terms, a circle is 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
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 book might also refer to something called the "standard"
form of the circle equaion. Unfortunately, "standard form" has
no fixed meaning that I've been able to determine, so you'll have to keep
track of what your book 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 recognize this formula; you will be expected to be able to read information from it.
The y2 term means the same thing as (y – 0)2, so the equation is really (x – 2)2 + (y – 0)2 = 52, and the center must be at (h, k) = (2, 0). Clearly, the radius is r = 5.
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.