
Skills to Master
·
Relate the
general equation for a circle to the Pythagorean Theorem
·
Determine the
center of a circle given its equation in standard form
·
Determine the
equation of a circle given the center and radius
·
Determine the
equation of a circle given the graph of the circle
·
Given the center
and radius of a circle, write the equation in general form
The Circle
A circle is a conic
section defined by the set of all points that are a constant distance (the
radius) from a given point (the center).
The equation of a circle can be derived using the Pythagorean Theorem,
which states that in a right triangle, the sum of the squares of the legs of
the triangle is equal to the square of the hypotenuse. That is, if the length of the legs are a and b and if the length of the hypotenuse is c, then
.

In the above graph, notice that if the radius is r, then for any point (x, y) on the
circle,
. This is the general equation of a circle
centered at the origin.
What if the circle center is not at the origin?

In the above figure, the center is at (h, k) and the point (x, y) is any point on the circle. Notice that the length of the base of the
right triangle is found by subtracting
. Similarly, the length of the side of the
triangle is found by subtracting
. If the radius (the third side of the
triangle) is r, then by the
Pythagorean Theorem, the equation of the circle is
. This equation also holds for the first
example because the values of h and k are each zero.
Example
Determine the coordinates of each circle’s center and radius.
Center: ( ), radius:
Center: ( ), radius:
Center: ( ), radius:
Center: ( ), radius:
Center: ( ), radius:
If given the graph of a circle, determine its equation by reading
the values of h, k, and r from the graph.
Example

What are the coordinate of the center in the above graph? (
, )
What is the radius of the circle?
The equation of the circle above is:
The general form of the equation of a circle is
. To convert a circle from standard form (as
above) to the general form, simply use FOIL to multiply the two squared
binomials, subtract the value of the radius from both sides, and combine like
terms.
Example
Convert the given equations from standard form to general form.
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Example
Find the equation of the circle with the given properties. Write your answer in general form.
Center at (0, 0), radius = ![]()
Center at (-3, 2), radius = 5
Center at (-1, -2), radius = ![]()