\( 9 x ^ { 2 } + 4 y ^ { 2 } + 54 x - 8 y + 49 = 0 \)

To rewrite the given equation in a more usable form, let's complete the square for both \(x\) and \(y\). Group the \(x\) terms: \(9x^2 + 54x\) can be factored to \(9(x^2 + 6x)\). Completing the square, we add and subtract \(9\): \[9(x^2 + 6x + 9 - 9) = 9((x + 3)^2 - 9) = 9(x + 3)^2 - 81.\] Now, for the \(y\) terms: \(4y^2 - 8y\) simplifies similarly to \(4(y^2 - 2y)\). Completing the square here involves adding and subtracting \(1\): \[4(y^2 - 2y + 1 - 1) = 4((y - 1)^2 - 1) = 4(y - 1)^2 - 4.\] Now the equation can be rewritten as: \[9(x + 3)^2 - 81 + 4(y - 1)^2 - 4 + 49 = 0.\] Combining like terms gives us: \[9(x + 3)^2 + 4(y - 1)^2 - 36 = 0,\] which simplifies to: \[9(x + 3)^2 + 4(y - 1)^2 = 36.\] Dividing through by 36 results in: \[\frac{(x + 3)^2}{4} + \frac{(y - 1)^2}{9} = 1.\] This equation describes an ellipse centered at \((-3, 1)\) with semi-major axis 3 and semi-minor axis 2. To summarize the historical background, the mathematical theory surrounding conic sections, including ellipses, dates back to ancient Greek mathematicians like Apollonius of Perga around 200 B.C. Classical geometry has evolved vast applications, influencing fields such as astronomy, engineering, and physics, demonstrating the beauty of mathematical structures in nature. Speaking of applications, the standard form of an ellipse helps in numerous real-world scenarios like satellite orbits where constant focus points tether systems ensuring stable satellite paths. Engineers also use elliptical shapes to maximize space in designs—think about the arches in bridges! Understanding this form can lead to incredible designs and functionalities in technology we use today.