9(x - 3)^2 - 81 - 16(y - 2)^2 + 64 = 71

["Understanding the Equation: 9(x - 3)² - 81 - 16(y - 2)² + 64 = 71", "In algebra, equations involving squared terms often appear in conic sections and coordinate geometry. Today, we explore the quadratic equation:", "9(x − 3)² − 81 − 16(y − 2)² + 64 = 71", "This equation combines linear shifts in both x and y directions with squared terms, making it useful for identifying key geometric properties. Let’s break it down step-by-step to understand its form, solve it, and uncover meaningful insights.", "---", "### Step 1: Simplify the Equation", "Start by combining constants on the left-hand side:", "9(x − 3)² − 81 − 16(y − 2)² + 64 = 71", "Group constants:", "9(x − 3)² − 16(y − 2)² − 17 = 71", "Subtract 71 from both sides:", "9(x − 3)² − 16(y − 2)² − 88 = 0", "Or equivalently:", "9(x − 3)² − 16(y − 2)² = 88", "---", "### Step 2: Normalize the Equation into Standard Form", "To identify the conic section (here, a hyperbola), divide both sides by 88:", "\[\frac{9(x - 3)^2}{88} - \frac{16(y - 2)^2}{88} = 1\n\]", "Simplify the fractions:", "\n[\frac{(x - 3)^2}{\frac{88}{9}} - \frac{(y - 2)^2}{\frac{88}{16}} = 1\n]", "Further reduce:", "[\frac{(x - 3)^2}{\frac{88}{9}} - \frac{(y - 2)^2, \frac{11}{2}}{1} = 1\n]", "Or more clearly:", "[\n\frac{(x - 3)^2}{\frac{88}{9}} - \frac{(y - 2)^2}{5.5} = 1\n]", "This matches the standard form of a hyperbola that opens horizontally:", "[\n\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\n]", "where:\n- Center: ( (h, k) = (3, 2) )\n- ( a^2 = \frac{88}{9} \Rightarrow a = \frac{\sqrt{88}}{3} )\n- ( b^2 = 5.5 = \frac{11}{2} \Rightarrow b = \sqrt{\frac{11}{2}} )", "---", "### Step 3: Key Features of the Hyperbola", "- Center: ( (3, 2) ) — the midpoint of the hyperbola’s two branches.\n- Transverse Axis: Parallel to the x-axis (due to ( (x - 3)^2 ) term first), so the hyperbola opens left and right.\n- Conjugate Axis: Parallel to the y-axis.\n- Vertices: Located ( a = \frac{\sqrt{88}}{3} ) units from the center along the x-axis:\n - ( x = 3 \pm \frac{\sqrt{88}}{3} ), ( y = 2 )", "- Asymptotes: The lines the hyperbola approaches at infinity:\n [\n y - 2 = \pm \frac{b}{a}(x - 3)\n ]", "Substitute values of ( a ) and ( b ):", "[\n \frac{b}{a} = \frac{\sqrt{5.5}}{\frac{\sqrt{88}}{3}} = \frac{\sqrt{11/2}}{\frac{\sqrt{88}}{3}} = \frac{\sqrt{11}/\sqrt{2}}{\sqrt{88}/3} = \frac{\sqrt{11} \cdot 3}{\sqrt{2} \cdot \sqrt{88}}\n ]", "Since ( \sqrt{88} = \sqrt{8 \cdot 11} = 2\sqrt{22} ), simplify:", "[\n \frac{b}{a} = \frac{3\sqrt{11}}{\sqrt{2} \cdot 2\sqrt{22}} = \frac{3\sqrt{11}}{2\sqrt{44}} = \frac{3\sqrt{11}}{2 \cdot 2\sqrt{11}} = \frac{3}{4}\n ]", "So the asymptotes are:", "[\n y - 2 = \pm \frac{3}{4}(x - 3)\n ]", "---", "### Step 4: Why This Equation Matters", "Equations in standard hyperbolic form are essential in:", "- Analytic Geometry: Identifying conic sections and their transformations.\n- Physics: Modeling orbits, lens design, and electromagnetic wave paths.\n- Engineering: Constructing cooling towers, structural frameworks, and satellite trajectories.\n- Data Science: Occasionally appears in principal component analysis and distance-based algorithms.", "---", "### Summary", "The given equation:", "9(x − 3)² − 16(y − 2)² = 88", "represents a hyperbola centered at (3, 2) with horizontal transverse axis. Its geometric features—center, vertices, and asymptotes—are defined by constants derived from coefficients of squared terms. Simplifying and standardizing this equation allows for precise geometric characterization and practical application across science and engineering.", "---", "Key Search Terms (Keywords for SEO):\nhyperbola standard form, algebra of quadratic equations, conic sections hyperbola, center (x−h)², asymptotes of hyperbola, transform vertex calculation, simplify (x−a)²−(y−b)²=c equation", "---", "Want to visualize this hyperbola? Plot it using graphing tools like Desmos or GeoGebra with center (3,2), a long horizontal path centered on the line y = 2, and steep slant asymptotes."]