9x^2 - 54x - 16y^2 + 64y = 71
["# Solving the Conic Section: 9x² – 54x – 16y² + 64y = 71", "Understanding and interpreting conic sections is fundamental in algebra and coordinate geometry. The equation ( 9x^2 - 54x - 16y^2 + 64y = 71 ) represents a hyperbola—a familiar but often misunderstood curve. In this SEO-optimized article, we’ll walk you through step-by-step how to rewrite the equation into standard form, analyze its properties, and explore real-world applications.", "---", "## What Type of Conic Section is This?", "The given equation is:\n[ 9x^2 - 54x - 16y^2 + 64y = 71 ]", "- The coefficient of ( x^2 ) is +9 (positive)\n- The coefficient of ( y^2 ) is –16 (negative)", "This sign contrast confirms it is a hyperbola.", "---", "## Step-by-Step Completing the Square", "To rewrite the equation in standard form, we complete the square for both ( x ) and ( y ).", "### Group terms by variable:\n[\n(9x^2 - 54x) + (-16y^2 + 64y) = 71\n]", "### Factor out coefficients of squared terms:\n[\n9(x^2 - 6x) - 16(y^2 - 4y) = 71\n]", "### Complete the square:", "- For ( x^2 - 6x ):\n Take half of –6: ((-3)^2 = 9)\n Add and subtract 9 inside the parentheses.\n Note: Because factored out a 9, adding 9 inside affects the expression by ( 9 \ imes 9 = 81 ) overall.", "- For ( y^2 - 4y ):\n Half of –4 is –2, square is 4\n Add and subtract 4 inside, but factored with –16, so subtract ( -16 \ imes 4 = -64 ) overall.", "[\n9(x^2 - 6x + 9 - 9) - 16(y^2 - 4y + 4 - 4) = 71\n]\n[\n9[(x - 3)^2 - 9] - 16[(y - 2)^2 - 4] = 71\n]", "### Expand and simplify:\n[\n9(x - 3)^2 - 81 - 16(y - 2)^2 + 64 = 71\n]\n[\n9(x - 3)^2 - 16(y - 2)^2 - 17 = 71\n]\n[\n9(x - 3)^2 - 16(y - 2)^2 = 88\n]", "### Divide both sides by 88:\n[\n\frac{9(x - 3)^2}{88} - \frac{16(y - 2)^2}{88} = 1\n]", "Simplify:\n[\n\frac{(x - 3)^2}{\frac{88}{9}} - \frac{(y - 2)^2}{\frac{88}{16}} = 1\n]\n[\n\frac{(x - 3)^2}{\frac{88}{9}} - \frac{(y - 2)^2}{5.5} = 1\n]", "This is the standard form:\n[\n\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\n]\nindicating a horizontal hyperbola centered at ( (h, k) = (3, 2) ).", "---", "## Key Properties of the Hyperbola", "| Property | Value |\n|------------------------|-------------------------------|\n| Center | ( (3, 2) ) |\n| Transverse Axis | Horizontal (along x-direction)|\n| ( a^2 = \frac{88}{9} ) | ( a = \frac{\sqrt{88}}{3} ) |\n| ( b^2 = 5.5 ) | ( b = \sqrt{5.5} ) |\n| Asymptotes | ( y - 2 = \pm \frac{b}{a}(x - 3) ) ( y - 2 = \pm \frac{\sqrt{5.5}}{\sqrt{88}/3}(x - 3) )\n ( y - 2 = \pm \frac{3\sqrt{5.5}}{\sqrt{88}}(x - 3) ) |", "---", "## Graphing the Hyperbola", "- Center: Plot point ( (3, 2) )\n- Vertices: ( (3 \pm a, 2) = \left(3 \pm \frac{\sqrt{88}}{3}, 2\right) )\n- Asymptotes: These guide the curve’s branches and are linear approximations:\n ( y = 2 \pm \frac{b}{a}(x - 3) )", "---", "## Real-World Applications", "Hyperbolas appear in:", "- Physics: Trajectories of certain projectiles under relativistic effects\n- Astronomy: Some comet orbits follow hyperbolic paths\n- Navigation: LORAN and GPS triangulation use hyperbolic principles\n- Engineering: Cooling towers, concert hall designs, and optical mirrors exploit hyperbolic shapes for structural and functional stability", "---", "## Summary", "The equation ( 9x^2 - 54x - 16y^2 + 64y = 71 ) simplifies to a standard hyperbola centered at ( (3, 2) ) with horizontal transverse axis. Understanding completing the square and identifying conic types enhances problem-solving in applied and theoretical math.", "---", "## FAQs", "Q: Why does the equation represent a hyperbola?\nA: The ( x^2 ) and ( y^2 ) terms have opposite signs, confirming hyperbolic behavior.", "Q: How do I graph this hyperbola?\nA: Identify center ( (3,2) ), plot vertices by computing ( a = \sqrt{88}/3 ), and draw asymptotes using slope ( \pm b/a ).", "Q: Can this equation model real physical phenomena?\nA: Yes—hyperbolas describe certain light paths, motion in non-uniform fields, and structural engineering designs.", "---", "Keywords: hyperbola equation, completing the square, conic sections, 9x² - 54x - 16y² + 64y = 71, hyperbola standard form, asymptotes, center, transverse axis, real-world applications.", "---", "Optimized for search: This guide teaches you how to rewrite and analyze the hyperbola equation ( 9x^2 - 54x - 16y^2 + 64y = 71 ), step-by-step, explaining key steps, properties, and applications—perfect for students, educators, and math enthusiasts. Master conic sections with clarity and confidence!"]
