Question

How to find the center and radius of `x^2+y^2-6x-10y=-9`?

Answer 1

The centre is `(3,5)` and the radius is 5

Explanation:

We know that the general form of a circle looks something like this:
`(x-h)^2+(y-k)^2=r^2`
where `(h,k)` is the centre and r is the radius

`x^2+y^2-6x-10y=-9` can be solved by completing the square

`(x^2-6x)+(y^2-10y)=-9`

`(x^2-6x+9)+(y^2-10y+25)=-9+9+25`

`(x-3)^2+(y-5)^2=25`

Therefore, the centre is `(3,5)` and the radius is 5

Answer 2

`"centre "=(3,5)" and radius "=5`

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"`
`"the radius"`

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

`x^2-6x+y^2-10y=-9`

`x^2+2(-3)x color(red)(+9)+y^2+2(-5)ycolor(magenta)(+25)=-9color(red)(+9)color(magenta)(+25)`

`(x-3)^2+(y-5)^2=25larrcolor(blue)"in standard form"`

`"centre "=(3,5)" and "r=sqrt25=5`