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x² + y² – 6x + 8y = 144

The equation of a circle in the xy-plane is shown above. 

What is the diameter of the circle?

Let d = diameter of the circle

Let r = radius of the circle

Let d = 2r

$\begin{array}{|rcll|} \hline x^2 + y^2 -6x+8y &=& 144\\ (x^2-6x) + (y^2+8y) &=& 144 \\ \left(x-\frac{6}{2}\right)^2 -\left(\frac{6}{2}\right)^2 +\left(y+\frac{8}{2}\right)^2 -\left(\frac{8}{2}\right)^2 &=& 144 \\ \left(x-3\right)^2 - 3^2 +\left(y+4\right)^2 -4^2 &=& 144 \\ \left(x-3\right)^2 - 9 +\left(y+4\right)^2 -16 &=& 144 \\ \left(x-3\right)^2 +\left(y+4\right)^2 &=& \underbrace{144+16 + 9}_{=r^2}\\\\ r^2 &=& 144+16 + 9 \\ r^2 &=& 169 \\ r &=& 13 \quad & | \quad \cdot 2 \\ 2r &=& 26 \quad & | \quad d = 2r \\ \mathbf{d} &\mathbf{=}& \mathbf{26} \\ \hline \end{array}$

The diameter of the circle is 26.

The center of the circle is ( 3, -4 ).