Question: Find the center of the hyperbola $9x^2 - 18x - 16y^2 + 64y = 144$.

["SEO Optimized Article: Finding the Center of the Hyperbola $9x^2 - 18x - 16y^2 + 64y = 144$", "Understanding the geometric properties of conic sections like hyperbolas is essential in algebra, geometry, and advanced mathematics. One fundamental task when analyzing a hyperbola is determining its center, which serves as the midpoint between the two branches and defines its positional anchor in the coordinate plane. In this article, we’ll walk through step-by-step how to find the center of the hyperbola defined by the equation:", "$$\n9x^2 - 18x - 16y^2 + 64y = 144\n$$", "---", "### What is a Hyperbola?", "A hyperbola is a type of conic section characterized by two distinct curved branches that open along a transverse axis. Unlike ellipses, hyperbolas have a center — a central point equidistant from both branches — and asymptotes that guide the intersecting lines the hyperbola approaches. The standard form of a hyperbola reveals key features, such as the center, vertices, and axes lengths.", "---", "### Step-by-Step: Rewriting the Equation to Find the Center", "To locate the center, we must rewrite the given equation into its standard form by completing the square for both $x$ and $y$ terms. Start with the original equation:", "$$\n9x^2 - 18x - 16y^2 + 64y = 144\n$$", "Step 1: Group $x$ and $y$ terms:", "$$\n(9x^2 - 18x) - (16y^2 - 64y) = 144\n$$", "Step 2: Factor out the coefficients of $x^2$ and $y^2$:", "$$\n9(x^2 - 2x) - 16(y^2 - 4y) = 144\n$$", "Step 3: Complete the square inside each parenthesis:", "- For $x^2 - 2x$: Take half of $-2$, square it to get $1$; add and subtract $1$ inside the group.\n Since it’s multiplied by 9, add $9 \cdot 1 = 9$ inside and compensate outside.", "- For $y^2 - 4y$: Half of $-4$ is $-2$, square is $4$. Add $4$ inside and subtract $16 \cdot 4 = 64$ outside.", "$$\n9(x^2 - 2x + 1 - 1) - 16(y^2 - 4y + 4 - 4) = 144\n$$", "$$\n9((x - 1)^2 - 1) - 16((y - 2)^2 - 4) = 144\n$$", "Step 4: Expand and simplify:", "$$\n9(x - 1)^2 - 9 - 16(y - 2)^2 + 64 = 144\n$$", "$$\n9(x - 1)^2 - 16(y - 2)^2 + 55 = 144\n$$", "$$\n9(x - 1)^2 - 16(y - 2)^2 = 89\n$$", "Step 5: Divide both sides by 89 to normalize:", "$$\n\frac{9(x - 1)^2}{89} - \frac{16(y - 2)^2}{89} = 1\n$$", "Or equivalently:", "$$\n\frac{(x - 1)^2}{\frac{89}{9}} - \frac{(y - 2)^2}{\frac{89}{16}} = 1\n$$", "---", "### The Standard Form and the Center", "This equation is now in the standard hyberbola form:", "$$\n\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\n$$", "where $(h, k)$ is the center of the hyperbola.", "From our completed square form, we see:", "- $h = 1$\n- $k = 2$", "Thus, the center of the hyperbola is at:", "$$\n\boxed{(1,\ 2)}\n$$", "---", "### Why the Center Matters", "Identifying the center is crucial because:", "- It serves as the coordinate reference point for plotting the hyperbola.\n- It defines symmetry axes: the transverse axis runs horizontally through $x = 1$, and the conjugate axis vertically through $y = 2$.\n- It helps compute vertices, foci, and asymptotes, which are essential for graphing and analyzing the hyperbola’s behavior.", "---", "### Summary", "- The given hyperbola equation was converted from general quadratic form to standard form via completing the square.\n- The center $(h, k) = (1, 2)$ was extracted directly from the squared terms.\n- Understanding this process empowers students and learners to solve similar conic section problems confidently.", "Key takeaway: To find the center of a hyperbola from its general equation, complete the square for both $x$ and $y$, then identify the coordinates $(h, k)$ appearing in the transformed $(x - h)^2$ and $(y - k)^2$ terms.", "---", "### Additional SEO Keywords\nOptimize this article further by including terms like:\n- Hyperbola center formula\n- Completing the square for hyperbola\n- Standard form conic sections\n- Finding hyperbola center\n- Conic sections guide\n- Algebra 2 conic sections", "---", "Final Thought: Mastering how to locate the center of a hyperbola transforms abstract equations into visual, spatially intuitive graphs — a cornerstone of graphing conic sections with precision.", "---\nKeywords: hyperbola center, how to find hyperbola center, standard form hyperbola, completing the square, conic sections, algebra 2, hyperbola analysis\nRelated searches: hyperbola vertex formula, hyperbola asymptotes, center vs foci hyperbola"]