Lesson Plan

Define an ellipse as the locus of points where the sum of distances from two foci equals a constant 2a, and derive the standard equation x^2/a^2 + y^2/b^2 = 1 where b^2 = a^2 - c^2

Define a hyperbola as the locus of points where the absolute difference of distances from two foci equals a constant 2a, and derive the standard equation x^2/a^2 - y^2/b^2 = 1 where b^2 = c^2 - a^2

Identify the key features of ellipses and hyperbolas from their equations: center, vertices, foci, axes, and (for hyperbolas) asymptotes

Compute the eccentricity of an ellipse or hyperbola and explain how it describes the shape

Distinguish between ellipses and hyperbolas based on their equations and geometric definitions

HSG.GPE.A.3

HSG.GPE.A.3: (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.