Question: Find the center of the hyperbola $ 4x^2 - 12x - 9y^2 + 18y = 27 $.

Question: Find the center of the hyperbola $ 4x^2 - 12x - 9y^2 + 18y = 27 $.

["Finding the Center of the Hyperbola: A Step-by-Step Guide", "When studying conic sections, one key task is identifying the center of hyperbolas—essential for graphing and understanding their geometric properties. The hyperbola defined by the equation\n[ 4x^2 - 12x - 9y^2 + 18y = 27 ]\nmight appear complex, but with the right method, locating its center becomes straightforward.", "---", "### What Determines the Center of a Hyperbola?", "For any hyperbola in standard form, the center lies at the point where the asymptotes intersect, or equivalently, at the midpoint of the transverse axis between the two real vertices. To reveal this center from a general quadratic equation, we must complete the square for both (x) and (y) terms.", "---", "### Step 1: Group (x) and (y) Terms", "Start by rearranging the equation:\n[ 4x^2 - 12x - 9y^2 + 18y = 27 ]", "Group the (x)-terms and (y)-terms:\n[ (4x^2 - 12x) - (9y^2 - 18y) = 27 ]", "---", "### Step 2: Factor Out Coefficients of Squared Terms", "Factor out the numerical coefficients:\n[ 4(x^2 - 3x) - 9(y^2 - 2y) = 27 ]", "---", "### Step 3: Complete the Square", "Complete the square inside each parentheses:", "- For (x^2 - 3x):\n ((x - \frac{3}{2})^2 - \left(\frac{3}{2}\right)^2 = (x - \frac{3}{2})^2 - \frac{9}{4})", "- For (y^2 - 2y):\n ((y - 1)^2 - 1)", "Substituting back:\n[ 4\left[(x - \frac{3}{2})^2 - \frac{9}{4}\right] - 9\left[(y - 1)^2 - 1\right] = 27 ]", "---", "### Step 4: Expand and Simplify", "Distribute the constants:\n[ 4(x - \frac{3}{2})^2 - 9 - 9(y - 1)^2 + 9 = 27 ]", "Simplify constants:\n[ 4(x - \frac{3}{2})^2 - 9(y - 1)^2 = 27 ]", "---", "### Step 5: Identify the Standard Form", "This equation is now in standard form:\n[ \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 ]\nwhere ((h, k)) is the center.", "Comparing:\n- ( h = \frac{3}{2} )\n- ( k = 1 )", "---", "### The Center of the Hyperbola", "Therefore, the center of the hyperbola\n[ 4x^2 - 12x - 9y^2 + 18y = 27 ]\nis located at the point\n[ \left( \frac{3}{2},\ 1 \right) ]", "---", "Why This Matters\nKnowing the center helps in graphing the hyperbola, analyzing its asymptotes, and solving related problems in physics, engineering, and coordinate geometry. Whether you're a student learning conic sections or a professional working with geometric models, mastering the completion of the square is essential.", "For quick reference, always complete the square on both (x) and (y) terms, then isolate the 1 on one side to identify the central coordinates.", "---", "Keywords: hyperbola center, find center hyperbola, completing the square, conic sections, hyperbola graphing, center of hyperbola equation, conic section tutorial, coordinate geometry"]

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