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Questions and Answers
什么是双曲线的定义?
什么是双曲线的定义?
什么是双曲线的焦点?
什么是双曲线的焦点?
什么是双曲线的軌跡?
什么是双曲线的軌跡?
双曲线的对称性是什么样的?
双曲线的对称性是什么样的?
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双曲线的渐近线是什么?
双曲线的渐近线是什么?
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双曲线的eccentricity是什么?
双曲线的eccentricity是什么?
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Study Notes
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
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