# What kind of conic is defined by the equation 2x^2+2y^2-8x+12y+2=0?

Dec 24, 2016

That is the equation of a circumference.

#### Explanation:

A circle is a set of points on the plane defined by the distance of any of them to a point $C \left(a , b\right)$ given is constant. At this point $C$ we will call it center of the circumference and the distance of this one to the points of the circumference receives the name of radius.

If a point $P \left(x , y\right)$ of the plane belongs to the circumference, its distance from the center $C \left(a , b\right)$ will be equal to the radius of the circle, $r$. The distance between $P$ and $C$ will be given by the module of $\vec{C P}$:

$d \left(P , C\right) = | \vec{C P} | = \sqrt{{\left(x - a\right)}^{2} + {\left(y - b\right)}^{2}}$,

which means that we will have:

$\sqrt{{\left(x - a\right)}^{2} + {\left(y - b\right)}^{2}} = r$.

If we raise everything to the square (to eliminate the root) and we develop the squares to eliminate the parentheses, we obtain:

${\left(\sqrt{{\left(x - a\right)}^{2} + {\left(y - b\right)}^{2}}\right)}^{2} = {r}^{2} \Rightarrow {\left(x - a\right)}^{2} + {\left(y - b\right)}^{2} = {r}^{2}$

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

Rearranging terms:

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

we obtain a generic formula for any circumference, which we can write of the form:

${x}^{2} + {y}^{2} + m x + n y + p = 0$,

being:

$m = - 2 a$,

$n = - 2 b$, and

$p = {a}^{2} + {b}^{2} - {r}^{2}$.

So, if we write the given equation in the following way:

$2 {x}^{2} + 2 {y}^{2} - 8 x + 12 y + 2 = 0 \Rightarrow {x}^{2} + {y}^{2} - 4 x + 6 y + 1 = 0$

(dividing everything by 2), it is clear that we have the equation of a circumference with:

m = - 4; n = 6; and p = 1.

Given that:

$- 2 a = - 4 \Rightarrow a = 2$,
$- 2 b = 6 \Rightarrow b = - 3$, and
${a}^{2} + {b}^{2} - {r}^{2} = 1 \Rightarrow {\left(2\right)}^{2} + {\left(- 3\right)}^{2} - {r}^{2} = 1 \Rightarrow r = 2 \sqrt{3}$,

it is clear that the proposed equation corresponds to a circumference of radius $r = 2 \sqrt{3}$ and center at the point $\left(2 , - 3\right)$.