Hyperbola Properties

Definition

• A hyperbola is a conic section defined as the set of all points in a plane such that the difference of the distances from two fixed points (foci) is constant.

Equations

• Standard equation: (x^2/a^2) - (y^2/b^2) = 1
• Center at origin: (x^2/a^2) - (y^2/b^2) = 1
• Center at (h, k): ((x-h)^2/a^2) - ((y-k)^2/b^2) = 1

Properties

• Foci: Two fixed points on the transverse axis, equidistant from the center of the hyperbola.
• Vertices: The points where the hyperbola intersects the transverse axis.
• Transverse axis: The axis that passes through the foci.
• Conjugate axis: The axis that passes through the center and is perpendicular to the transverse axis.
• Asymptotes: The lines that the hyperbola approaches as x or y increases without bound.
• Eccentricity: The ratio of the distance between the foci to the distance between the vertices.

Graphical Characteristics

• Open shape: A hyperbola is an open shape, extending to infinity in both directions.
• Symmetry: Hyperbolas have mirror symmetry about the transverse and conjugate axes.
• Branches: A hyperbola has two branches that open in opposite directions.

双曲线性质

• 双曲线是指所有点到两个固定点（焦点）的距离之差保持不变的平面内所有点的集合。

双曲线方程

• 标准方程： (x^2/a^2) - (y^2/b^2) = 1
• 原点为中心： (x^2/a^2) - (y^2/b^2) = 1
• 中心在 (h, k)： ((x-h)^2/a^2) - ((y-k)^2/b^2) = 1

双曲线性质

焦点

• 两个固定点，位于 ngang轴上，距双曲线中心等距

端点

• 双曲线与 ngang轴的交点

ngang轴

• 经过焦点的轴

共轭轴

• 经过中心，垂直于 ngang轴的轴

###渐近线

• 双曲线趋于无穷大的时候，接近的两条直线

离心率

• 焦点之间的距离与端点之间的距离之比

图形特征

• 开放形状：双曲线是一条开放的曲线，伸展到无限远
• 对称性：双曲线关于 ngang轴和共轭轴具有镜像对称性
• 分支：双曲线有两个朝向相反方向的分支