Question

Write the equation x2 + y2 + 6x + 8y + 24 = 0 in vertex form.

Answer 1

ANSWER

$(x+3)^2+(y+4)^2=1$

EXPLANATION

We want to write the equation of the circle in vertex form:

$x^2+y^2+6x+8y+24=0$

The first step is to group the x terms and y terms together and take the constant to the right-hand side of the equality sign:

$x^2+6x+y^2+8y=-24$

Now, complete the square for the x terms:

$\begin{gathered} x^2+6x+(\frac{6}{2})^2+y^2+8y=-24+(\frac{6}{2})^2 \\ x^2+6x+9+y^2+8y=-24+9 \\ (x+3)^2+y^2+8y=-15 \end{gathered}$

Repeat the process for the y terms:

$\begin{gathered} (x+3)^2+y^2+8y+(\frac{8}{2})^2=-15+(\frac{8}{2})^2 \\ (x+3)^2+y^2+8y+16=-15+16 \\ (x+3)^2+(y+4)^2=1 \end{gathered}$

That is the equation of the circle in vertex form.