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Learning Goal
Part of: Translate between the geometric description and the equation for a conic section — 3 of 3 cluster items
Derive conic equations
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.
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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.
What you'll learn
- 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
Prerequisites
Slides
Interactive presentations perfect for visual learners • 2 slide decks
Slide Video
Watch narrated slides play like a video lesson • Narrated slide playback