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{ \rv{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{\rv{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{\rv{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{\rv{2, 1}}# and radius #\green{r} = 1#.
#r=# #\sqrt{5}#
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{1, 8}# and radius #r=\sqrt{5}#.
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