Question

What is the center and radius of the circle with equation `x^2 + y^2 + 6x + 16y + 64 = 0`?

Answer 1

center: `(-3,-8)`
radius: `3`

Explanation:

Recall that the general equation of a circle is:

`color(blue)(|bar(ul(color(white)(a/a)(x-h)^2+(y-k)^2=r^2color(white)(a/a)|)))`

where:
`x=`x-coordinate
`h=`x-coordinate of circle's centre
`y=`y-coordinate
`k=`y-coordinate of circle's centre
`r=`radius of circle

Given the equation,

`x^2+y^2+6x+16y+64=0`

Subtract `64` from both sides.

`x^2+y^2+6x+16y=-64`

Grouping and bracketing the terms with `x` and `y` separately,

`x^2+6x+y^2+16y=-64`

`(x^2+6x)+(y^2+16y)=-64`

Complete the square within each bracket.

`(x^2+6xcolor(white)(i)color(red)(+(6/2)^2))+(y^2+16ycolor(white)(i)color(red)(+(16/2)^2))=-64`

`(x^2+6xcolor(white)(i)color(red)(+9))+(y^2+16ycolor(white)(i)color(red)(+64))=-64`

Add `9` and `64` to the right side of the equation.

`(x^2+6xcolor(white)(i)color(darkorange)(+9))+(y^2+16ycolor(white)(i)color(darkorange)(+64))=-64color(white)(i)color(darkorange)(+64)color(white)(i)color(darkorange)(+9)`

`(x^2+6x+9)+(y^2+16y+64)=9`

Rewrite each bracketed expression in factored form.

`(x+3)^2+(y+8)^2=9`

According to the general equation of a circle, the centre and radius can be determined by looking at the equation for the circle as,

`(x-(color(red)(-3)))^2+(y-(color(blue)(-8)))^2=color(purple)3^2`

`ul("Reminder": (x-color(red)h)^2+(y-color(blue)k)^2=color(purple)r^2)`

Since the centre is `(color(red)h,color(blue)k)` and the radius is `color(purple)r`, using the derived equation, we can determine that the centre of the circle is `(color(red)(-3),color(blue)(-8))` and the radius is `color(purple)3`.