Question

How do you write the equation of the circle in standard form, identify the center and radius of `x^2+y^2-2x+6y+9=0`?

Answer 1

`(x-1)^2+(y+3)^2=1`

Explanation:

`"the equation of a circle in standard form is "`

`color(red)(bar(ul(|color(white)(2/2)color(black)((x-a)^2+(y-b)^2=r^2)color(white)(2/2)|)))`

`"where "(a,b)" are the coordinates of the centre and r"`
`"is the radius"`

`"to obtain this form use "color(blue)"completing the square"`
`"on both x and y terms"`

`x^2-2x+y^2+6y+9=0`

`rArr(x^2+2(-1)xcolor(red)(+1)color(red)(-1))+(y^2+2(3)ycolor(blue)(+9)color(blue)(-9))+9=0`

`rArr(x-1)^2+(y+3)^2=0-9color(blue)(+9)color(red)(+1)`

`rArr(x-1)^2+(y+3)^2=1larrcolor(red)"in standard form"`

`rArr"centre "=(1,-3)" and radius "=1`