Equation of the circumference II: general equation

A circumference with center $C = (a, b)$ and radius $r$ can be rewritten in light of the reduced equation as:

$(x - a)^{2} + (y - b)^{2} = r^{2}$

Developing the squares of the above mentioned equation we obtain:

$x^{2} + y^{2} - 2 a x - 2 b y + a^{2} + b^{2} - r^{2} = 0$

and doing the change $A = - 2 a, B = - 2 b, C = a^{2} + b^{2} - r^{2}$ in:

$x^{2} + y^{2} - 2 a x - 2 b y + a^{2} + b^{2} - r^{2} = 0$

the new equation is obtained:

$x^{2} + y^{2} + A x + B y + C = 0$

This way we have found another analytical expression that defines the points of a circumference. This is the general equation of the circumference.

Let's see how to determine the radius and the center of a circumference from the general equation.

We can do the following:

$A = - 2 a, B = - 2 b, C = a^{2} + b^{2} - r^{2}$

We isolate these expressions in terms of $a$ , $b$ and $r$ . We have:

$a = - \frac{A}{2}$ $b = - \frac{B}{2}$ $r^{2} = a^{2} + b^{2} - C = (- \frac{A}{2})^{2} + (- \frac{B}{2})^{2} - C = \frac{A^{2} + B^{2} - 4 C}{4}$

And since we know that, in the limited expression, $(a, b)$ is the center and $r$ the radius, given a general equation:

$x^{2} + y^{2} + A x + B x + C = 0$

the center of such a circumference is the point $(- \frac{A}{2}, - \frac{B}{2})$ and the radius is $r = \sqrt{\frac{A^{2} + B^{2} - 4 C}{4}}$ .

Example

Let's suppose that they give us the circumference: $x^{2} + y^{2} - 2 x + 4 y - 4 = 0$ then we see that it is centred at the point:

$(- \frac{A}{2}, - \frac{B}{2}) = (- \frac{- 2}{2}, - \frac{4}{2}) = (1, - 2)$

and has radius:

$r = \sqrt{\frac{A^{2} + B^{2} - 4 C}{4}} = \sqrt{\frac{(- 2)^{2} + 4^{2} - 4 \cdot (- 4)}{4}} =$

$= \sqrt{\frac{4 + 16 + 16}{4}} = \sqrt{\frac{36}{4}} = \frac{6}{2} = 3$

Let's now see the inverse process, that is to say:

Example

Giving the general equation of the circumference that has, for example, radius $4$ and center $(- 5, 6)$ .

We write the reduced equation:

$(x - a)^{2} + (y - b)^{2} = r^{2} \Rightarrow (x + 5)^{2} + (y - 6)^{2} = 4^{2}$

developing the squares we have:

$(x + 5)^{2} + (y - 6)^{2} = 4^{2} \Rightarrow x^{2} + 10 x + 25 + y^{2} - 12 y + 36 = 16$

If we rearrange it and add all the independent terms, we obtain the general equation of the above mentioned circumference, which is:

$x^{2} + 10 x + 25 + y^{2} - 12 y + 36 = 16$

$x^{2} + y^{2} + 10 x - 12 y + 25 + 36 = 16$

$x^{2} + y^{2} + 10 x - 12 y + 45 = 0$

Let's see what happens when the circumference is centred on the origin and we want to write its general equation:

Since $(0, 0)$ is the center we have: $a = 0$ and $b = 0$ for which reason,

$\begin{matrix} 0 = a = - \frac{A}{2} \\ 0 = b = - \frac{B}{2} \end{matrix}} ⟹ {\begin{matrix} A = 0 \\ B = 0 \end{matrix}$

So that in the general equation, only quadratic terms and independent terms will exist, that is to say:

$x^{2} + y^{2} + C = 0$

Moving the constant term to the other side we obtain:

$x^{2} + y^{2} = - C$

where we know that:

$C = a^{2} + b^{2} - r^{2} = - r^{2}$

since the supposed center was $(0, 0)$ .

Note that for the circunference centered at the origin both equations are very similar.

Let's see an example:

Example

Circumference centred on the origin and radius $7$ .

Reduced equation: $x^{2} + y^{2} = 7^{2}$

General equation: $x^{2} + y^{2} + C = 0$ where $C = - 7^{2} ⟹ x^{2} + y^{2} - 7^{2} = 0$

Summing up we have:

Considering the circumference: $(x - a)^{2} + (y - b)^{2} = r^{2}$

Then the center is the point of the plane $(a, b)$ and the radius is $r$ .

Example

$(x - 8)^{2} + (y + 3)^{2} = 1$ has center at $(8, - 3)$ and radius $1$ .

Considering the circumference: $x^{2} + y^{2} + A x + B y + C = 0$

Then the center is at the point of the plane $(- \frac{A}{2}, - \frac{B}{2})$ and the radius is $r = \sqrt{(\frac{A}{2})^{2} + (\frac{B}{2})^{2} - C}$

Example

$x^{2} + y^{2} + x - 5 y - 2 = 0$ has center $(\frac{- 1}{2}, \frac{5}{2})$ and radius

$r = \sqrt{(\frac{1}{2})^{2} + (\frac{- 5}{2})^{2} - (- 2)} = \sqrt{\frac{1 + 25 + 8}{4}} = \sqrt{\frac{34}{4}} = \sqrt{\frac{17}{2}}$