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THE CIRCLE A circle is the locus of all points, in a
plane that is a fixed distance from a fixed point, called the center. The fixed distance spoken of here is the radius of the
circle. The equation of a circle with its center at the origin
(fig. 25) is
where (x,y) is a point on the circle and r is the
radius (r replaces d in the standard distance formula). Then
or
Figure
25.Circle with center at the origin. If the center of a circle, figure 26, is at some point
where x = h, y = k, then the distance of the point (x,y) from the center will
be constant and equal to _{
} or
Figure
26.Circle with center at (h,k). Equations (2.1) and (2.2) are the standard forms for the
equation of a circle. Equation (2.1) is merely a
special case of equation (2.2) in which h and k are equal to zero. The equation of a circle may also be expressed in the
general form:
where B, C, and D are
constants. Theorem: An equation of the second degree in which the
coefficients of the x^{2 }and y^{2 }terms
are equal and the xy term does not exist, represents a circle. Whenever we find an equation in the form of equation
(2.3), we should convert it to the form of equation (2.2) so that we have the
coordinates of the center of the circle and the radius as part of the equation.
This may be done as shown in the following example problems: EXAMPLE: Find the coordinates of the center and the radius
of the circle described by the following equation:
SOLUTION: First rearrange the terms
and complete the square in both x and y. Completing the
square is discussed in the chapter on quadratic solutions in Mathematics,
Volume 1. The procedure consists basically of adding certain quantities
to both sides of a seconddegree equation to form the sum of two perfect
squares. When both the first and seconddegree members are known, the square
of onehalf the coefficient of the firstdegree term is added to both sides of
the equation. This will allow the quadratic equation to be factored into the
sum of two perfect squares. To complete the square in x in the given equation
add the square of onehalf the coefficient of x to both
sides of the equation
then
completes the square in x. If we do the same for y,
completes the square in y. Transpose all constant terms to the righthand side and
simplify:
The equation is now in the standard form of equation
(2.2). This equation represents a circle with the center at (2,3) and with a
radius equal to
or 4. EXAMPLE:
Find the coordinates of the center and the
radius of the circle given by the equation ^{} SOLUTION:
Rearrange and complete the square in both x
and y:
Transposing all constant terms to the righthand side and
adding, results in
Reducing to the equation in standard form results in
