Circle centered at C(h,k)

# Circle centered at C(h, k)
## Analytic Geometry
### Arturo Sánchez González
### <a href="mailto:arturo.sanchez@upaep.mx" class="email">arturo.sanchez@upaep.mx</a>
### August 2021

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# Quick review of the canonical equation of a circle centered at the origin

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# Equation of a circle centered at the origin

The canonical equation of a circle of radius `$r$` centered at the origin is given by

`$$x^2 + y^2 = r^2$$`
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# What happen when the center of the circle is not at the origin?

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Let's consider a circle of radius `$\color{yellow}{r>0}$` centered at `$\color{cyan}{C(h,k)}$`.

Then, if `$\color{magenta}{P(x,y)}$` lies on that circle, `$\color{magenta}{P}$` **satisfies the definition of the circle**.

This means that
$$
d(\color{magenta}{P},\color{cyan}{C}) = \color{yellow}{r}
$$
And, again, using the **distance formula** we get the equality
$$
\sqrt{(\color{magenta}{x}-\color{cyan}{h})^2 + (\color{magenta}{y}-\color{cyan}{k})^2} = \color{yellow}{r}
$$
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.pull-left[
 And, taking squares in both sides we get
$$
(\color{magenta}{x}-\color{cyan}{h})^2 + (\color{magenta}{y}-\color{cyan}{k})^2 = \color{yellow}{r}^2
$$
This equation is called
.center[
STANDARD EQUATION OF THE CIRCLE WITH RADIUS `$\color{yellow}{r}$` CENTERED AT `$\color{magenta}{C(h,k)}$`
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# Homework

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Problem A

Let's consider a circle with center at `$C(-3,2)$` and radius `$11$`. Make a draw of this circle, and determine its canonical equation and its general equation

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Problem B

A pizza delivery area can be represented by a circle, and extend to the points `$(0,18)$` and `$(-6,8)$` (these points are on the **diameter** of this circle). Write two equations for the circle that models this delivery area.

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Problem C

Determine the center and the radius, and the sketch the circle with equation

$$
3x^2+3y^2−12x+4=0$$`