Circle Equations

A circle is a two-dimensional shape whose points are all equidistant from the center.

All circles with centers located at the origin (0, 0) have the equation:

x² + y² = r²
Where r is the radius of the circle.

So, a circle whose center is located at the origin (0, 0) with a radius of 4 has the equation:
x² + y² = 16

Standard Form of a Circle's Equation

A circle whose center is located at point (a, b) having a radius of length r has the equation:
(x -a)² + (y -b)² = r²
This is called the standard form of a circle's equation.       (This is sometimes called the center-radius form).

A circle's center coordinates and radius can be easily determined from the standard form.

A circle's center would be found at point   (a, b)   and its radius equals the   square root of r².

So, for example, a circle with the equation     (x -2)² + (y -1)² = 25

Would have its center located at   (2, 1) and   would have a radius equal to the   square root of 25 or 5.

General Form of a Circle's Equation

Multiplying out a circle's standard form:   (x -a)² + (y -b)² = r²

produces the general form of a circle's equation:  x² + y² + cx + dy + e = 0

Figure 1 below shows a circle whose center is at point (4, 3) with a radius of 2.
How do we find the standard form equation of this circle?
We put the center coordinates (4, 3) into the a and b variables and we square the radius for r²

(x -4)² + (y -3)² = 4
By multiplying this out, we get the general form
x² -8x + 16 +y² -6y + 9 = 4
x² +y² -8x -6y +21 = 0

Going From General Form To Standard Form

How do you convert a general form equation into a standard form equation?

1) Let's take this general form equation as an example.

x² +y² +16x -18y -145 = 0

2) Set the equation up in this manner and move the "non-x" term to the right.
(x  )² + (y  )² = 145

3) Looking at the equation in step (1), take the x coefficient (16)  and the y coefficient (-18)  
  divide both these numbers by two, giving us (8, -9)
  Insert these numbers into equation 2, paying attention to the signs.
(x + 8)² + (y - 9)² = 145

4) Finally, take the numbers from step 3 (8, -9), square them (64, 81) then add both these numbers to 145, giving us the finished equation:
  (x + 8)² + (y - 9)² = 290

A calculator that computes a circle's equation after 3 points have been input is located here.

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