Equations of Ellipses and Hyperbolas
Aligned to G-GPE.A.3 — Common Core State Standards for Mathematics.
What this lesson teaches
An ellipse is the set of points whose distances to two foci have a constant sum; a hyperbola is where the difference of those distances is constant. Applying the distance formula and simplifying gives standard forms: ellipse x^2/a^2 + y^2/b^2 = 1, hyperbola x^2/a^2 - y^2/b^2 = 1.
Worked example
Set up an ellipse with foci (-4, 0) and (4, 0) and distance sum 10. For a point (x, y): sqrt((x+4)^2 + y^2) + sqrt((x-4)^2 + y^2) = 10. Here 2a = 10 so a = 5, and c = 4 (focal distance). Use b^2 = a^2 - c^2 = 25 - 16 = 9. The equation is x^2/25 + y^2/9 = 1.
Practice questions
- An ellipse has foci at (-3,0) and (3,0) with distance sum 10. Find a, c, and b^2.
- Write the standard equation of the ellipse from the previous problem.
- A hyperbola has a = 3 and c = 5. Find b^2 using c^2 = a^2 + b^2 and write x^2/a^2 - y^2/b^2 = 1.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Ecuaciones de elipses e hipérbolas
Una elipse es el conjunto de puntos cuya suma de distancias a dos focos es constante; una hipérbola es donde la diferencia de esas distancias es constante. Aplicar la fórmula de distancia y simplificar da las formas estándar: elipse x^2/a^2 + y^2/b^2 = 1, hipérbola x^2/a^2 - y^2/b^2 = 1.
Ejemplo: Plantea una elipse con focos (-4, 0) y (4, 0) y suma de distancias 10. Para un punto (x, y): raíz((x+4)^2 + y^2) + raíz((x-4)^2 + y^2) = 10. Aquí 2a = 10, así que a = 5, y c = 4 (distancia focal). Usa b^2 = a^2 - c^2 = 25 - 16 = 9. La ecuación es x^2/25 + y^2/9 = 1.
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