Curves of the Second Order

Study axis-aligned second-order curves from their general equations, including classification by the sign of AC, reduction to canonical form, and finding the centre or vertex, foci, eccentricity, directrices, and asymptotes.

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BASIC DEFINITIONS
Basic Definitions
The definitions, canonical equations, parameters, and construction rules for the ellipse, hyperbola, and parabola: the tools used throughout the problems below.
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1Problem 1 · Investigating a Second-Order EquationInvestigate the equation 4x2+9y2−8x−36y+4=04x^2+9y^2-8x-36y+4=0.Start →2Problem 2 · Investigating a Second-Order EquationInvestigate the equation x2+y2−2x+4y+5=0x^2+y^2-2x+4y+5=0.3Problem 3 · Investigating a Second-Order EquationInvestigate the equation x2+y2+4x+5=0x^2+y^2+4x+5=0.4Problem 4 · Investigating a Second-Order EquationInvestigate the equation x2−9y2+2x+36y−44=0x^2-9y^2+2x+36y-44=0.5Problem 5 · Investigating a Second-Order EquationInvestigate the equation 16y2−9x2−32y+18x+7=016y^2-9x^2-32y+18x+7=0.