Different descriptions of a circle

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.

Expanded

It is important to note that when we expand the brackets it is no longer 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 $\green{\sqrt{25}} = \green{5}$

Completing the square.

We can rewrite an equation of the form $x^2 + ax$ to $\left(x + \frac{a}{2}\right)^2 - \left(\frac{a}{2}\right)^2$ . This technique is called completing the square.

Example

$x^2 - 8x = (x- 4)^2 -16$

$y^2 + 14y - 3 = (y+7)^2 - 49 - 3$

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 $\green{r} = 1$ .

A circle $c$ is given by $c: \left(x-7\right)^2+\left(y-5\right)^2=2$
What is the center $M$ and the radius $r$ of circle $c$ ?
Enter the coordinates of $M$ as $\rv{a,b}$ with the correct values for $a$ and $b$ .

$M=$ $\rv{7, 5}$
$r=$ $\sqrt{2}$

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, $M= \rv{7, 5}$ and radius $r=\sqrt{2}$ .

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