4(x - 4)^2 - 9(y - 3)^2 + 17 = -115 \implies 4(x - 4)^2 - 9(y - 3)^2 = -132.

["# Understanding the Equation: ( 4(x - 4)^2 - 9(y - 3)^2 = -132 )", "The equation ( 4(x - 4)^2 - 9(y - 3)^2 = -132 ) represents a powerful algebraic form that reveals important geometric and algebraic properties. This article explains how to interpret, simplify, and analyze this equation, particularly focusing on its role in identifying conic sections, simplifying expressions, and solving related problems.", "---", "## What This Equation Represents", "The given equation is:", "[\n4(x - 4)^2 - 9(y - 3)^2 = -132\n]", "This form resembles the standard structure of a degenerate or non-standard conic section, derived from a second-degree equation in two variables. Specifically, it involves squared terms centered at ((4, 3)) with different coefficients and mixed linear shifts ((x - 4)) and ((y - 3)).", "Because the left-hand side combines positive and negative squared terms equal to a negative constant, this typically suggests a hyperbola-like identity, though constrained by constant scaling. Importantly, this equation is commonly reduced by dividing through by the constant on the right to standardize the conic form.", "---", "## Rewriting the Equation in Standard Form", "To better understand the geometry, divide both sides by (-132):", "[\n\frac{4(x - 4)^2}{-132} - \frac{9(y - 3)^2}{-132} = 1\n]", "Simplify the fractions:", "[\n-\frac{(x - 4)^2}{33} + \frac{(y - 3)^2}{\frac{132}{9}} = 1 \quad \Rightarrow \quad \frac{(y - 3)^2}{\frac{44}{3}} - \frac{(x - 4)^2}{33} = 1\n]", "This matches the standard hyperbola form:", "[\n\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1\n]", "where:\n- Center: ((h, k) = (4, 3))\n- (a^2 = \frac{44}{3}) ⇒ (a = \sqrt{\frac{44}{3}})\n- (b^2 = 33) ⇒ (b = \sqrt{33})\n- (a^2 / b^2 = \frac{44/3}{33} = \frac{4}{9}), confirming a vertical transverse axis hyperbola.", "---", "## Geometric Insight: Transformed Hyperbola", "This equation describes a hyperbola centered at ((4, 3)), vertically oriented, meaning it opens up and down. Unlike standard axis-aligned hyperbolas, the terms here reflect shifted coordinate axes:", "- The (x)-term subtracts 4, indicating horizontal shift\n- The (y)-term adds and subtracts 3, indicating vertical shift", "Because the signs are opposite and the right-hand side is negative (after standardization), this is consistent with a hyperbola that extends vertically.", "The minimal value of the left-hand side occurs at the center, ((4, 3)):", "[\n4(0)^2 - 9(0)^2 = 0\n]", "But in the scaled form, the right-hand side is (-132), so the expression reaches (-132) along its curve, representing the transverse axis extent defined by the hyperbola’s vertices and asymptotes.", "---", "## Solving for Key Features", "### Vertices\nFor vertical hyperbolas of the form (\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1), the vertices lie along the vertical axis at:", "[\n(h, k \pm a)\n]", "Here:", "[\na = \sqrt{\frac{44}{3}} \approx 3.83\n]", "So the vertices are at:", "[\n(4, 3 \pm \sqrt{44/3}) = \left(4, 3 \pm \frac{2\sqrt{33}}{3}\right)\n]", "---", "### Asymptotes\nThe asymptotes of such hyperbolas are given by:", "[\ny - k = \pm \frac{a}{b}(x - h)\n]", "Substitute values:", "[\ny - 3 = \pm \frac{\sqrt{44/3}}{\sqrt{33}} (x - 4) = \pm \sqrt{\frac{44}{99}} (x - 4) = \pm \frac{2\sqrt{33}}{33}(x - 4)\n]", "Simplified slope:\n[\n\pm \frac{2}{3}\sqrt{\frac{11}{3}} \approx \pm 0.8019\n]", "These lines define the long-term direction the hyperbola approaches.", "---", "## Practical Applications", "Equations of this form appear in optimization, physics modeling, and coordinate transformations. Because of their shifted centers, they model phenomena with specific pivot points, such as:", "- Structural stress distribution around a focal point\n- Reflection properties in optical systems\n- Trajectories in relative motion problems", "The algebra supporting simplification enables easier substitution, graphing, and parametric analysis—essential for both theoretical and applied math.", "---", "## Conclusion", "The equation ( 4(x - 4)^2 - 9(y - 3)^2 = -132 ) is far more than symbolic—its fully simplified hyperbolic form reveals a centered, vertically oriented hyperbola with vertices at ((4, 3 \pm \sqrt{44/3})) and asymptotes sloping at ( \pm \frac{2\sqrt{11}}{3\sqrt{3}} ).", "By recognizing its standard structure and manipulating constants, we unlock deep geometric understanding and practical utility. Mastering such equations strengthens skills in conic sections, coordinate geometry, and algebraic manipulation—cornerstones of advanced mathematics.", "---", "Keywords: (4(x - 4)^2 - 9(y - 3)^2 = -132), hyperbola equation, center ((4, 3)), standard form, asymptotes, conic sections, geometry, algebraic manipulation, shifted hyperbola", "---", "Related Topics:\n- How to graph (4(x - 4)^2 - 9(y - 3)^2 = -132)\n- Difference between hyperbola and ellipse equations\n- Transforming coordinates of conic sections\n- Applications of hyperbolas in engineering and physics", "---", " debts to algebraic geometry and coordinate transformations."]