Substitute back: $ 9(x - 2)^2 - 36 - 4(y - 3)^2 + 36 = 36 $. Simplify: $ 9(x (x - 2)^2 - 4) - 4(y - 3)^2 = 36 $. Divide by 36: $ \frac{(x - 2)^2}{4} - \frac{(y - 3)^2}{9} = 1 $. The center is at $ (2, 3) $. The center is $\boxed{(2, 3)}$.

["Simplifying and Interpreting the Conic Equation: The Hyperbola Centered at (2, 3)", "Understanding conic sections often begins with recognizing their standard forms, which reveal key properties like center, orientation, and shape. One such transformation leads us to a hyperbola defined by the equation:", "$$\n9(x - 2)^2 - 36 - 4(y - 3)^2 + 36 = 36\n$$", "At first glance, this equation may appear complex, but through careful simplification, we uncover its geometric essence. Let's walk through the step-by-step process and explain why its center is clearly $\boxed{(2, 3)}$.", "---", "### Step 1: Simplify the Left-Hand Side", "Begin by combining like terms:", "$$\n9(x - 2)^2 - 36 - 4(y - 3)^2 + 36 = 36\n$$", "Notice that $-36 + 36$ cancels out, leaving:", "$$\n9(x - 2)^2 - 4(y - 3)^2 = 36\n$$", "---", "### Step 2: Normalize to Standard Form", "To identify the conic type, divide every term by 36:", "$$\n\frac{9(x - 2)^2}{36} - \frac{4(y - 3)^2}{36} = 1\n$$", "Simplify the fractions:", "$$\n\frac{(x - 2)^2}{4} - \frac{(y - 3)^2}{9} = 1\n$$", "---", "### Step 3: Recognize the Standard Hyperbola Form", "The simplified equation matches the standard form of a hyperbola centered at $(h, k)$:", "$$\n\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\n$$", "Here, comparing:\n- $ h = 2 $\n- $ k = 3 $", "Thus, the center of the hyperbola is at $ \boxed{(2, 3)} $.", "---", "### Step 4: Interpret the Structure", "The simplified equation:", "$$\n\frac{(x - 2)^2}{4} - \frac{(y - 3)^2}{9} = 1\n$$", "tells us this:", "- The transverse axis is horizontal because the $x$-term is positive.\n- $a^2 = 4 \Rightarrow a = 2$, indicating the vertices are 2 units left and right from $x = 2$, at $(0, 3)$ and $(4, 3)$.\n- $b^2 = 9 \Rightarrow b = 3$, related to the conjugate axis length.", "---", "### Why This Format Matters", "By completing the algebraic "substitution" and dividing properly, we transform the original expression into its canonical form. This reveals the hyperbola’s core geometric properties — notably, its center — simplifying analysis, graphing, and applications in modeling curved trajectories, optimization, and physical phenomena.", "---", "### Summary", "- Original equation: $9(x - 2)^2 - 4(y - 3)^2 = 36$\n- Normalized: $\frac{(x - 2)^2}{4} - \frac{(y - 3)^2}{9} = 1$\n- Center: $ \boxed{(2, 3)} $\n- Hyperbola oriented horizontally with $a = 2$, $b = 3$", "Understanding this transformation empowers you to interpret and solve more complex conic problems with confidence.", "---", "Optimize your math skills: Always simplify equations carefully—precision reveals clarity. From $x$ and $y$ terms to hyperbola centers, every step brings you closer to mastering coordinate geometry."]