Equation of a Circle
The equation of a circle is a fundamental concept in geometry that describes a circle as the set of all points equidistant from a fixed point called the center. This distance is known as the radius. In the context of analytic geometry, the equation can be expressed in two main forms: the standard form and the general form. The standard form is derived from the distance formula and outlines the relationship between the coordinates of the center and any point on the circle. The general form, on the other hand, results from algebraic manipulation of the standard form and allows for a more versatile representation of the circle's equation.
By employing these algebraic equations, one can utilize calculus to explore various properties of circles, such as determining the slope of tangent lines at given points. The analytical approach enables a deeper understanding of geometric problems and introduces powerful techniques that may be challenging to apply without the equation. Overall, the equation of a circle serves as an essential tool in both mathematics education and advanced geometric analysis.
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Equation of a Circle
In the roughly 5,500 years since the invention of the wheel, circles have been among the most important of geometric figures. Around 300 BC, in his classic geometry text Elements, Euclid of Alexandria wrote, "A circle is a plane figure contained by a single line [which is called a circumference], (such that) all of the straight-lines radiating towards [the circumference] from one point amongst those lying inside the figure are equal to one another. And the point is called the center of the circle." This definition of a circle is fine for the purposes of Euclidean geometry, which is still widely taught today. However it has one major drawback: It does not lend itself well to re-interpretation as an equation. The study of geometric figures through algebraic equations is called analytic geometry, and it is in this framework that the equation of a circle is most readily obtained.
Overview
A circle is the set of all points
that are equidistant from a fixed point
, called the center. The common distance between
and each of the points
is called the radius
. See Figure 1.
The key word in our definition of a circle is equidistant. Every point on the circle is the same distance from the center. The distance between two points
and
is the given by the distance formula :
. See Figure 2.
Squaring both sides of the distance formula yields the following.
Let
be at the center
of a circle, let
be at a point
on the same circle, and let the distance between them be the radius
of the circle. See Figure 3.
Making the substitutions indicated in Figure 3 into equation
, the equation of a circle is obtained in factored form.
Equation
is also known as the standard form of the equation of a circle.
The equation of a circle need not be given in standard form. The equation can also appear in non-factored form, also known as general form, which is given below.
In equation
,
and
are constants. The general form is obtained from the standard form by squaring the binomials on the left side of
and collecting like terms. See Table 1, for example:
The standard form can be recovered from the general form by completing the square in both
and
in
. See Table 2, for example:
One great advantage of the analytic approach is that the powerful techniques of calculus can be brought to bear on geometric problems. For instance, it can be shown using calculus that the slope
of the line tangent to the circle described by
is given in terms of
and
as follows.
As a concrete example, the slope of the line that is tangent to the circle given by
at the point
is given by
as follows.
Such a result is far more difficult to obtain without an algebraic equation of a circle.
Bibliography
Clark, David M. Euclidean. Geometry: A Guided Inquiry Approach. Berkeley: American Mathematical Soc., 2012.
Larson, Ron, and Bruce H. Edwards. Calculus. Boston: Cengage, 2014.
Larson, Ron, and David C. Falvo. Precalculus. Boston: Cengage, 2014.
Ostermann, Alexander, and Gerhard Wanner. Geometry by Its History. New York: Springer, 2012.
Swokowski, Earl W., and Jeffery A. Cole. Algebra and Trigonometry with Analytic Geometry. Belmont, CA: Brooks, 2008.