Find properties of hyperbolas from equations in general form

Learning Objectives Convert the general form of a hyperbola's equation to its standard form. Identify the center (h, k) and orientation of a hyperbola from its equation. Calculate the coordinates of the vertices and foci. Derive the equations of the asymptotes. Distinguish between the general forms of hyperbolas, ellipses, and parabolas. Use the derived properties to sketch a graph of the hyperbola. Ever wondered about the shape of a sonic boom's shockwave or the path of a comet slingshotting past the sun? ☄️ Those are hyperbolas, and today we'll learn to decode their secrets from complex equations! We will start with a jumbled equation called the 'general form' and learn the powerful algebraic technique of 'completing the square' to trans...

TermDefinitionExample General Form of a Conic SectionThe equation Ax² + Cy² + Dx + Ey + F = 0, where A, C, D, E, and F are constants. For a hyperbola, the coefficients A and C must have opposite signs (one positive, one negative).9x² - 16y² - 36x - 32y - 124 = 0 is the general form of a hyperbola because the x² coefficient (9) is positive and the y² coefficient (-16) is negative. Standard Form (Horizontal)The form \frac{(x-h)²}{a²} - \frac{(y-k)²}{b²} = 1. This represents a hyperbola that opens left and right, with a horizontal transverse axis.\frac{(x-2)²}{9} - \frac{(y+1)²}{16} = 1 Standard Form (Vertical)The form \frac{(y-k)²}{a²} - \frac{(x-h)²}{b²} = 1. This represents a hyperbola that opens up and down, with a vertical transverse axis.\frac{(y-3)²}{25} - \frac{(x-1)²}{4} = 1 Center...

Converting from General to Standard Form The method of 'Completing the Square'. For a term like z² + bz, add (b/2)² to create a perfect square trinomial: z² + bz + (b/2)² = (z + b/2)². This is the fundamental algebraic process used to transform the general form equation. You must group x-terms and y-terms, factor out leading coefficients, complete the square for both variables, and balance the equation before dividing to make the right side equal to 1. Hyperbola Foci Relationship c² = a² + b² Use this formula to find the distance 'c' from the center to each focus. Remember that for a hyperbola, 'a²' is always the denominator of the positive term in the standard form equation. Asymptote Equations For a horizontal hyperbola: y - k = ±\frac{b...