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 examples
Example 1
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.
Example 2
Set up a hyperbola with foci (0, -5) and (0, 5) and difference of distances 8. The foci lie on the y-axis, so the hyperbola opens up and down: y²/a² - x²/b² = 1. Here 2a = 8 gives a = 4, and c = 5. For a hyperbola b² = c² - a² = 25 - 16 = 9. The equation is y²/16 - x²/9 = 1.
Example 3
Common mistake: using the same b formula for both curves. An ellipse uses b² = a² - c² because the foci are INSIDE it and c < a; a hyperbola uses b² = c² - a² because the foci are OUTSIDE the branches and c > a. Swapping them gives a negative square. Check the order of c and a first: foci inside means ellipse, foci beyond the vertices means hyperbola.
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 1: 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.
Ejemplo 2: Plantea una hipérbola con focos (0, -5) y (0, 5) y diferencia de distancias 8. Los focos están sobre el eje y, así que la hipérbola abre hacia arriba y hacia abajo: y²/a² - x²/b² = 1. Aquí 2a = 8 da a = 4, y c = 5. Para una hipérbola, b² = c² - a² = 25 - 16 = 9. La ecuación es y²/16 - x²/9 = 1.
Ejemplo 3: Error común: usar la misma fórmula de b para ambas curvas. La elipse usa b² = a² - c² porque los focos están DENTRO y c < a; la hipérbola usa b² = c² - a² porque los focos quedan FUERA de las ramas y c > a. Intercambiarlas da un cuadrado negativo. Revisa primero el orden de c y a: focos dentro significa elipse, focos más allá de los vértices significa hipérbola.
More Expressing Geometric Properties with Equations lessons
Teach this lesson today
An AI tutor that guides with questions instead of just giving answers — all K–12 core subjects, every child in the family.
Start free — no card required