Question

How do you graph `x^2 + y^2 – 6x + 8y + 9 = 0`?

Answer 1

Draw a circle with radius `4` and center at `(3,-4)`

Explanation:

Given:
`color(white)("XXX")x^2+y^2-6x+8y+9=0`

Re-arrange into standard circle equation form:

Re-group:
`(x^2-6x)+(y^2+8y)=-9`

Complete the squares:
`(x^2-6xcolor(red)(+3^2))+(y^2+8ycolor(blue)(+4^2))=-9color(red)(+3^2)color(blue)(+4^2)`

Write as squared binomials and simplify the right side:
`(x-3)^2+(y+4)^2= 4^2`
or
`(x-color(green)(3))^2+(y-color(green)(color(white)("")(-4)))^2=color(brown)(4^2)`

Recalling that the standard circle equation is
`color(white)("XXX")(x-color(green)(a))^2+(y-color(green)(b))^2=color(brown(r)^2`
for a circle with center `(color(green)(a),color(green)(b))` and radius `color(brown)(r)`

The given equation is that of a circle with center `(color(green)(3),color(green)(color(white)("")(-4)))` and radius `color(brown)(4)`

graph{x^2+y^2-6x+8y+9=0 [-5.72, 12.066, -8.194, 0.69]}