Convert equations of conic sections from general to standard form

Learning Objectives Identify the type of conic section directly from its general form equation. Apply the 'completing the square' method to quadratic expressions involving both x and y variables. Systematically convert the general form equation of a conic section into its corresponding standard form. Correctly handle leading coefficients when completing the square. Extract key geometric properties (like center, vertices, foci, radius) from the derived standard form equation. Recognize and classify degenerate conic cases that may arise during the conversion process. How does a GPS receiver pinpoint your exact location on Earth using signals from satellites in orbit? 🛰️ The math behind it involves intersecting conic sections, and understanding their equations is the...

TermDefinitionExample General Form of a Conic SectionThe equation of any conic section can be written in the form Ax² + Cy² + Dx + Ey + F = 0, where A, C, D, E, and F are constants. For the conics studied in this course, we assume the Bxy term is zero.9x² + 16y² - 36x + 96y + 36 = 0 is the general form equation for an ellipse. Standard FormA specific format for the equation of a conic section that explicitly reveals its geometric properties, such as its center, vertices, and orientation.The standard form for a horizontal ellipse is (x-h)²/a² + (y-k)²/b² = 1, where (h, k) is the center. Completing the SquareAn algebraic technique used to convert a quadratic expression of the form x² + bx into a perfect square trinomial, (x + b/2)², by adding the value (b/2)².To complete the square for x² -...

The 'Completing the Square' Formula For an expression x² + bx, add (b/2)² to create a perfect square trinomial: x² + bx + (b/2)² = (x + b/2)² This is the core algebraic manipulation used in the conversion process. It must be applied to both the x-terms and y-terms. Remember to balance the equation by adding the same value to the other side. Standard Forms of Conic Sections Circle: (x-h)² + (y-k)² = r² \\ Parabola: (x-h)² = 4p(y-k) or (y-k)² = 4p(x-h) \\ Ellipse: \frac{(x-h)²}{a²} + \frac{(y-k)²}{b²} = 1 \\ Hyperbola: \frac{(x-h)²}{a²} - \frac{(y-k)²}{b²} = 1 or \frac{(y-k)²}{a²} - \frac{(x-h)²}{b²} = 1 These are the target forms for your conversion. The final arrangement of the squared terms, the signs, and the constant on the right side determine which conic you h...