Name: hyperbolas: foci, vertices, and asymptotes
Uploaded: 2025-11-26T05:44:06.000Z
Duration: 1 min 30 s
Channel: JoVE

当一个平面穿过圆锥的两个圆锥面时，会形成双曲线，产生两条开口曲线，称为分支。分支沿长度为 2a 的横轴延伸，其中 a 为从中心到每个顶点的距离。与此垂直的是共轭轴，其长度为 2b，定义一个尺寸为2的矩形a 2倍b其对角线向外延伸为渐近线，引导但永不与分支相交。双曲线定义为到两个固定点（称为焦点）的距离之差的绝对值为常数且等于 2a 的点的集合。焦点位于 x 轴上，分别在负 c 和正 c 处，其中 c 为从中心到每个焦点的距离。将点P与每个焦点之间的距离公式进行计算，得到的表达式在平方后可消除平方根。随后展开平方项，并进行代数化简。进一步平方并化简可消去剩余的根号。然后代入关系式 b² = c² - a² —— 这是勾股定理的一种形式 —— 即可得到标准方程。双曲面形状被用于冷却塔，因为这种形状能够增强结构强度和气流。

双曲线

Hyperbolas

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Education

教育

JoVE Core

Math Fundamentals

数学基础

Unit

单位

Analytic Geometry

解析几何

KEY TERMS AND CONCEPTS

关键术语与概念

hyperbolas: foci, vertices, and asymptotes

Hyperbolas are conic sections with two separate branches. They form when a plane cuts through a double-napped cone at a steeper angle than the cone’s slope, so the plane passes through both nappes. The two curves are mirror images and open away from each other along the transverse axis.

The closest points on the branches to the center are called vertices. The distance from the center to a vertex is labeled a. The axis perpendicular to the transverse axis is the conjugate axis, and it is linked to b. The value of b changes the shape of the branches, but it does not change how open they are.

A hyperbola can also be defined by distance. It is the set of all points where the absolute difference of the distances to two fixed points, called foci, stays constant. This property sets hyperbolas apart from ellipses and parabolas.

The standard equation of a hyperbola has squared terms with opposite signs. It may be written for a horizontal opening or for a vertical opening, with (h, k) as the center. The positive term shows the transverse axis, which is the direction the branches open. From this form, you can identify the center, vertices, and asymptotes.

Hyperbolic shapes also appear in engineering. Power plant cooling towers often use a hyperbolic contour. This shape can improve structural stability by spreading stress efficiently, and it can also support thermal performance by helping natural convection and airflow through the tower.

Hyperbolas: Foci, Vertices, and Asymptotes

inclination of a line

Slope Angle and Line Direction

The slope angle shows how a line tilts on the Cartesian plane. It gives the line's direction by linking geometry with slope, which tells how steeply the line rises or falls. A line has no thickness, but its orientation still matters. This angle is measured counterclockwise from the positive x-axis. For nonhorizontal lines, the inclination angle ranges from 0 to π radians. That range helps describe the line's tilt in a clear geometric way. When two lines intersect, each line has its own...

parabolas

Parabolas: Focus, Directrix, and Symmetry

A parabola is a conic section with a clear geometric rule. It forms when a plane cuts a double-napped cone parallel to the cone’s slant height. The result is an open curve used in coordinate geometry and defined by a fixed point and a fixed line. That fixed point is the focus, and the fixed line is the directrix. Every point on a parabola stays the same distance from the focus as from the directrix. In mathematics, this set of points is called the locus of points with equal distance from both.<br...

reflective property of parabolas

Parabola Focus and Signal Reflection

A parabola is a U-shaped conic section with an important reflective property. It forms when a plane cuts a double-napped cone in a direction parallel to one side of the cone. In geometry, a parabola is also defined as the set of points that are equally far from a fixed point, called the focus, and a fixed line, called the directrix. This curve has a useful way of handling light, sound, and other parallel rays. Incoming rays that travel parallel to the axis of symmetry reflect toward one point.

ellipses

Ellipse Geometry: Foci, Axes, and Equations

An ellipse is a closed curve in geometry with two foci, a major axis, and a minor axis. It is formed when a right circular cone is cut by an inclined plane that does not pass through the cone’s base. The shape is symmetric, which makes its structure easy to study in analytic geometry. One key way to define an ellipse is by distance. For any point on the curve, the sum of the distances to the two fixed points, called the foci, stays constant. That constant sum is equal to the length of the...

eccentricity of an ellipse

Ellipse Shape and Orbits Explained

An ellipse is a key conic section with two foci, or fixed points inside the shape. For any point on the ellipse, the sum of the distances to the two foci stays the same. A pencil, string, and two pins can show this shape in a simple way. If the string is held tight while the pencil moves, it traces an ellipse. The size and shape of an ellipse depend on its eccentricity, written as e. Eccentricity is the ratio of the distance from the center to a focus, c, to the semi-major axis length, a. When...

hyperbolas

Hyperbolas are conic sections with two separate branches. They form when a plane cuts through a double-napped cone at a steeper angle than the cone’s slope, so the plane passes through both nappes. The two curves are mirror images and open away from each other along the transverse axis. The closest points on the branches to the center are called vertices. The distance from the center to a vertex is labeled a. The axis perpendicular to the transverse axis is the conjugate axis, and it is linked...

geometry of hyperbolas

Hyperbola Properties and Asymptotes

A hyperbola is a curve defined by two fixed points called foci. For any point on the curve, the absolute difference in its distances to the foci stays constant. This distance rule gives the hyperbola its shape and separates it from other conic sections. The standard equation shows two branches that extend forever. Each branch moves closer to two asymptotes, which are lines that guide the curve as it grows. The parameters a and b control key features of the graph. The value a measures the...

polar equations of conics

Conic Sections in Polar Form

Conic sections can be written in polar form using a focus and a directrix. The focus is a fixed point, and the directrix is a fixed line. A point on the conic is defined by the ratio of its distance to the focus and its distance to the directrix. That ratio is called the eccentricity. This setup gives one framework for ellipses, parabolas, and hyperbolas. When the focus is placed at the origin of the polar coordinate system, one polar equation can represent any conic section. That makes the...

key terms and concepts

Explore hyperbolas through their foci, vertices, axes, asymptotes, and equations, plus a real-world example in power plant cooling towers.

Hyperbolas: Foci, Vertices, and Asymptotes | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Hyperbolas: Foci, Vertices, and Asymptotes

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