A circle can be defined as the locus of all points that satisfy the equation

and h,k are the coordinates of its center.

(x-h)^{2} + (y-k)^{2} = r^{2}

where r is the radius of the circle,
and h,k are the coordinates of its center.

Try this
Drag the point C and note how h and k change in the equation. Drag P and note how the radius squared changes in the equation.

Recall from Basic Equation of a Circle, that when the circle's center is at the origin, the formula is When the circle center is elsewhere, we need a more general form. We add two new variables h and k that are the coordinates of the circle center point:

(x-h)^{2} + (y-k)^{2} = r^{2}

We subtract these from x and y in the equation to translate ("move") the center back to the origin.
If you compare the two formulae, you will see that the only difference is that the h and k variables are subtracted from the x and y terms before squaring them:

Basic | (x)^{2} + (y)^{2} = r^{2} |

General | (x-h)^{2} + (y-k)^{2} = r^{2} |

When we see the equation of a circle such as we know it is a circle of radius 9 with its center at x = 3, y = –2.

- The radius is 9 because the formula has r
^{2}on the right side. 9 squared is 81. - The y coordinate is negative because the y term in the general equation is (y-k)
^{2}.

In the example, the equation has (y+2), so k must be negative: (y– (–2))^{2}becomes (y+2)^{2}.

The equation is then a little simpler. Since the center is at the origin, h and k are both zero. So the general form becomes which simplifies down to the basic form of the circle equation: For more on this see Basic Equation of a Circle.

Instead of using the Pythagorean Theorem to solve the right triangle in the circle above, we can also solve it using trigonometry. This produces the so-called parametric form of the circle equation as described in Parametric Equation of a Circle. This parametric form is especially useful in computer algorithms that draw circles and ellipses. It is described in An Algorithm for Drawing Circles.

- In the applet above, click 'reset' and 'hide details'.
- Drag the points C and P to create a new circle.
- Write the general formula for the resulting circle.
- Click on 'show details' to check your result.

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