A hyperbola is a conic section formed by all points whose distances from two fixed points have a constant difference. It appears as two separate branches that open in opposite directions. Hyperbolas matter in mathematics, navigation, astronomy, and physics because they model relationships involving inverse behavior, wave timing, and orbital paths.
On a coordinate plane, their shape is controlled by a center, foci, vertices, and asymptotes.
Key Facts
- For a horizontal hyperbola centered at (h, k): (x - h)^2/a^2 - (y - k)^2/b^2 = 1.
- For a vertical hyperbola centered at (h, k): (y - k)^2/a^2 - (x - h)^2/b^2 = 1.
- The foci of a horizontal hyperbola are (h - c, k) and (h + c, k), where c^2 = a^2 + b^2.
- The constant difference property is |d1 - d2| = 2a, where d1 and d2 are distances from a point on the hyperbola to the two foci.
- For a horizontal hyperbola, the asymptotes are y - k = ±(b/a)(x - h).
- The vertices of a horizontal hyperbola are (h - a, k) and (h + a, k).
Vocabulary
- Hyperbola
- A hyperbola is the set of all points in a plane where the absolute difference of distances to two fixed foci is constant.
- Focus
- A focus is one of the two fixed points used to define a hyperbola by distance difference.
- Center
- The center is the midpoint between the two foci and between the two vertices of a hyperbola.
- Vertex
- A vertex is a point where a branch of the hyperbola is closest to the center.
- Asymptote
- An asymptote is a line that the branches of a hyperbola approach but do not cross as they extend outward.
Common Mistakes to Avoid
- Using c^2 = a^2 - b^2 for a hyperbola is wrong because hyperbolas use c^2 = a^2 + b^2, unlike ellipses.
- Forgetting that the sign order determines direction is wrong because x^2 first means a horizontal hyperbola, while y^2 first means a vertical hyperbola.
- Drawing the branches through the foci is wrong because the branches pass through the vertices, and the foci lie inside the opening beyond the vertices.
- Treating asymptotes as part of the hyperbola is wrong because asymptotes are guide lines that the curve approaches, not points on the curve.
Practice Questions
- 1 For the hyperbola x^2/16 - y^2/9 = 1, find the center, vertices, foci, and asymptote equations.
- 2 A horizontal hyperbola has center (2, -1), a = 3, and b = 4. Write its standard equation and find the coordinates of its foci.
- 3 Explain how you can tell from the equation (y - 2)^2/25 - (x + 1)^2/9 = 1 whether the hyperbola opens up and down or left and right.