A home-schooled student is studying the properties of conic sections. Determine the type of conic section represented by the equation \( 9x^2 + 4y^2 - 36x + 16y + 36 = 0 \).

["Understanding Conic Sections: A Home-Homeschooled Student’s Guide to Identifying Conic Types\nHow to Analyze the Equation (9x^2 + 4y^2 - 36x + 16y + 36 = 0)", "As a home-schooled student diving into the fascinating world of conic sections, one essential skill is learning how to identify the type of conic described by a general second-degree equation. Today, we’ll explore a classic example: (9x^2 + 4y^2 - 36x + 16y + 36 = 0), determine which conic it represents, and understand the process step-by-step.", "---", "### What Are Conic Sections?", "Conic sections include circles, ellipses, parabolas, and hyperbolas—curves formed by slicing a cone with a plane under different angles. In algebra, they are represented by second-degree equations of the form:\n[\nAx^2 + Bxy + Cy^2 + Dx + Ey + F = 0\n]\nThe nature (type and orientation) of the conic depends on the coefficients (A), (B), and (C), especially the discriminant (B^2 - 4AC).", "---", "### Step 1: Rewrite the Equation into Standard Form", "Given:\n[\n9x^2 + 4y^2 - 36x + 16y + 36 = 0\n]\nGroup (x) and (y) terms:\n[\n(9x^2 - 36x) + (4y^2 + 16y) + 36 = 0\n]", "Factor out coefficients of squared terms:\n[\n9(x^2 - 4x) + 4(y^2 + 4y) + 36 = 0\n]", "Complete the square:", "For (x):\n(x^2 - 4x \rightarrow (x - 2)^2 - 4)\nFor (y):\n(y^2 + 4y \rightarrow (y + 2)^2 - 4)", "Substitute back:\n[\n9\left[(x - 2)^2 - 4\right] + 4\left[(y + 2)^2 - 4\right] + 36 = 0\n]\n[\n9(x - 2)^2 - 36 + 4(y + 2)^2 - 16 + 36 = 0\n]\n[\n9(x - 2)^2 + 4(y + 2)^2 - 16 = 0\n]\n[\n9(x - 2)^2 + 4(y + 2)^2 = 16\n]", "Divide entire equation by 16 to normalize:\n[\n\frac{9(x - 2)^2}{16} + \frac{4(y + 2)^2}{16} = 1\n]\n[\n\frac{(x - 2)^2}{\frac{16}{9}} + \frac{(y + 2)^2}{4} = 1\n]\n[\n\frac{(x - 2)^2}{(4/3)^2} + \frac{(y + 2)^2}{2^2} = 1\n]", "---", "### Step 2: Analyze the Standard Form", "This equation matches the standard form of an ellipse:\n[\n\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\n]\nwhere (a) and (b) are positive constants defining the semi-major and semi-minor axes.", "Here:\n- Center: ((2, -2))\n- Semi-major axis squared: (a^2 = \frac{16}{9} \Rightarrow a = \frac{4}{3})\n- Semi-minor axis squared: (b^2 = 4 \Rightarrow b = 2)\nSince (a^2 < b^2), the major axis is vertical.", "---", "### Step 3: Cross-Check Using the Discriminant", "Earlier, the general form was (Ax^2 + Bxy + Cy^2 + \dots = 0).\nHere, (A = 9), (C = 4), (B = 0).\nDiscriminant:\n[\n\Delta = B^2 - 4AC = 0^2 - 4(9)(4) = -144 < 0\n]\nA negative discriminant confirms the conic is an ellipse.", "---", "### Final Answer", "The equation (9x^2 + 4y^2 - 36x + 16y + 36 = 0) represents an ellipse located at ((2, -2)) with vertical major axis, semi-major axis length ( \frac{4}{3} ), and semi-minor axis length (2).", "---", "### Why This Matters for Home-Study Students", "Understanding how to classify conics by examining coefficients and completing the square builds a strong foundation in analytical geometry. As you continue studying, applying these techniques systematically will help you recognize and analyze conics confidently in advanced math and real-world applications—from optics to orbital mechanics.", "Keep practicing with different equations: complete the square, compute the discriminant, and match forms to master conic identification!", "---", "Keywords: conic sections, ellipse identification, conic analysis, completing the square, conic equations, home-schooled math, algebra, geometry, discriminant of conics"]