### Geometry: Circles

### Different descriptions of a circle

A circle is a geometric shape in the plane that is determined by a point #P#, the center of the circle, and a radius #r#. The circle consists of all points which have distance #r# from #P#. When we want to actually calculate with circles, it is useful to have an equation for a circle.

A circle with centre #\blue{P} =\blue{ (a, b)}# and radius #\green{r} > 0# can be described by the equation

\[(x - \blue {a} )^2 + (y-\blue{b})^2 = \green{r}^2\]

Such an equation is often called **the equation of a circle. **

It is not immediately apparent from an equation whether or not it can be rewritten to the equation of a circle. One can however try the technique of **completing the square**.

Completing the square

An equation of the form

\[x^2 + dx + y^2 + ey + f = 0\]

can be rewritten to

\[(x - \blue{a})^2 + (y- \blue{b} )^2 = c\]

Whenever #c# is positive it can be rewritten to an equation of a circle with centre #\blue{(a, b)}# and radius #\green{r} = \green{\sqrt{c}}#

\[(x - \blue{a})^2 + (y-\blue{b})^2 = \green{r}^2\]

**Example **

The equation

\[x^2 + y^2 + 2y - 8x - 8 =0\]

can be rewritten to \[(x - \blue{4})^2 -16 + (y -\blue{1})^2 -1 -8 =0\]

Bringing the constant terms to the right we get the equation

\[(x - \blue{4})^2 + (y-\blue{1})^2 = \green{5}^2\]

This is the equation of the circle with center #\blue{(4, 1)}# and radius #\green{\sqrt{25}} = \green{5}#

Sketching a circle

From the equation of a circle it is possible to sketch the circle using a pencil and compass. First draw the centre of the circle and then trace a circle with the right radius using the compass. Below is an example with centre #\blue{P} = \blue{(2, 1)}# and radius #\green{r} = 1#.

#r=# #1#

In a circle equation of the form #(x-a)^2+(x-b)^2=r^2# the center point #M# is equal to #\rv{a,b}# and the radius to #r#.

In this case, therefore, #M= \rv{-2, -6}# and radius #r=1#.

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