Question: A comet follows a hyperbolic trajectory described by $ 4x^2 - 9y^2 - 32x + 54y + 115 = 0 $. Determine the center of the hyperbola.

["Title: Discovering the Comet’s Path: How to Identify the Center of a Hyperbola from Its Equation", "Meta Description:\nUnlock the secrets of a comet’s hyperbolic orbit by analyzing its equation. Learn how to determine the center of the hyperbola described by $ 4x^2 - 9y^2 - 32x + 54y + 115 = 0 $. Step-by-step guide for astronomers and math enthusiasts alike.", "---", "### Introduction\nComets traveling through our solar system often follow precise yet complex paths — sometimes described by elegant conic sections. When a comet’s trajectory follows a hyperbolic path, understanding its geometry becomes key to predicting its behavior. In this article, we dive into the equation $ 4x^2 - 9y^2 - 32x + 54y + 115 = 0 $ and reveal the location of its center — a critical piece of information for both astronomers and mathematicians.", "---", "### Understanding the Hyperbola’s Equation", "The general form of a conic section is:\n$$\nAx^2 + By^2 + Dx + Ey + F = 0\n$$\nFor this equation to represent a hyperbola, the coefficients of $ x^2 $ and $ y^2 $ must have opposite signs — which they do here ($ +4 $ and $ -9 $), confirming a hyperbolic trajectory.", "To identify the hyperbola’s center, we must rewrite the equation in standard form by completing the square.", "---", "### Step-by-Step Completing the Square", "Start with the given equation:\n$$\n4x^2 - 9y^2 - 32x + 54y + 115 = 0\n$$", "Group $ x $-terms and $ y $-terms:\n$$\n(4x^2 - 32x) + (-9y^2 + 54y) + 115 = 0\n$$", "Factor out coefficients of squared terms:\n$$\n4(x^2 - 8x) - 9(y^2 - 6y) + 115 = 0\n$$", "Complete the square inside each parenthesis:\nFor $ x^2 - 8x $, add and subtract $ \left(\frac{8}{2}\right)^2 = 16 $:\n$$\nx^2 - 8x = (x - 4)^2 - 16\n$$", "For $ y^2 - 6y $, add and subtract $ \left(\frac{6}{2}\right)^2 = 9 $:\n$$\ny^2 - 6y = (y - 3)^2 - 9\n$$", "Substitute back:\n$$\n4\left[(x - 4)^2 - 16\right] - 9\left[(y - 3)^2 - 9\right] + 115 = 0\n$$", "Distribute coefficients:\n$$\n4(x - 4)^2 - 64 - 9(y - 3)^2 + 81 + 115 = 0\n$$", "Simplify constants:\n$$\n4(x - 4)^2 - 9(y - 3)^2 + 132 = 0\n$$", "Move constant to the right:\n$$\n4(x - 4)^2 - 9(y - 3)^2 = -132\n$$", "Divide both sides by $-132$ to normalize:\n$$\n\frac{(x - 4)^2}{-33} - \frac{(y - 3)^2}{-14.67} = 1\n$$\nRewriting with positive denominators (multiplying numerator and denominator by $-1$ appropriately):\n$$\n\frac{(y - 3)^2}{132/9} - \frac{(x - 4)^2}{132/4} = 1\n$$", "Thus, the standard form becomes:\n$$\n\frac{(y - 3)^2}{\frac{44}{3}} - \frac{(x - 4)^2}{33} = 1\n$$", "---", "### Identifying the Center", "The standard form of a hyperbola that opens vertically (since $ y^2 $ coefficient is positive) is:\n$$\n\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1\n$$\nHere, $ (h, k) $ is the center of the hyperbola.", "From our completed work, we see:\n- $ h = 4 $\n- $ k = 3 $", "Thus, the center of the hyperbola is at $ (4, 3) $.", "---", "### Why the Center Matters for Comets", "Knowing the center helps astronomers track the comet’s motion, calculate its closest approach to Earth (perigee), and predict future positions. The hyperbolic trajectory indicates the comet is on an escape path — likely from the outer solar system, possibly the Oort Cloud or Kuiper Belt.", "---", "### Final Thoughts", "By completing the square and analyzing the coefficients, we determined that the hyperbolic path of this comet has its center located at $ (4, 3) $. This precise geometric insight is vital for both theoretical astrophysics and practical comet observation.", "If you’re charting celestial motion or studying orbital mechanics, recognizing the conic section type and identifying its center unlocks deeper understanding — whether you're a student, researcher, or comet watcher.", "---", "Keywords: hyperbola center, comet trajectory, hyperbolic orbit equation, eliminate $(4, 3)$ hyperbola center, conic section hyperbola, completing the square, astronomy coordinates", "For more essential tools in celestial navigation, explore articles on conic sections, orbital mechanics, and astronomical observation techniques."]