Course Content
第1章：数学归纳法
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1 Mathematical Induction 数学归纳法 [练习题]
第2章：反三角函数
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2.2 Calculation of Inverse Trigonometric Functions
2.3 Inverse Trigonometry Function Identities 反三角函数的恒等式
2.4 Inverse Trigonometric Function Equations 反三角函数方程式
第3章：微分法（二）
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3.2 Derivative of Inverse Trigonometric Functions
3.3 Derivative of Logarithm Functions 对数函数的导数
3.4 Derivative of Exponential Functions 指数函数的导数
3.5 Logarithmic Differentiation 对数微分法
3.6 L’Hospital’s Rule 罗比达法则
第4章：坐标变换
第5 章 圆锥曲线
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5.2 Parabola 抛物线
5.3 Ellipse 椭圆
5.4 Hyperbola 双曲线
第7章：参数方程式
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7.3 Parametric Equations and Locus 参数方程式与轨迹
7.5 Parametric Equations of Conic 圆锥曲线的参数方程式
第9章 复数
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9.2 Complex Number and Its Evaluation复数及其运算
9.5 de Moivre’s Theorem 隶美佛定理
9.6 The nth Root of Complex Number 复数的n次方根
9.7 Roots of Unary nth Order Equations 一元n 次方程式的根
第10章 微分法的应用（二）
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10.1 Graphing for Curve 曲线的作图法
10.2 The Approximate Solution of Unary Equations 一元方程式的近似解
第12章 定积分及其应用（二）
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12.2 Calculation of Area under Polar Coordinate 极坐标系下的面积计算
第13章 常微分方程式
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13.2 First Order Differential Equations 一阶微分方程
13.2.2 First Order Homogeneous Differential Equations 一阶齐次微分方程
13.2.3 First Order Linear Differential Equations 一阶线性微分方程
13.3 Application of First Order Differential Equations 一阶常微分方程的应用
13.4 Second Order Linear Differential Equation with Constant Coefficient 二阶常系数线性微分方程
高三数学 | 高级数学
About Lesson
5.4 Hyperbola 双曲线

(1)

Find the centre, vertex, axis of symmetry, transverse semi-axis, conjugate semi-axis, eccentricity, focus, directrix, length of latus rectum and asymptotes for the following hyperbola and sketch the graph.

求下列各双曲线的中心, 顶点, 对称轴, 半实轴长, 半虚轴长, 离心率, 焦点, 准线, 通径长度和渐近线並描其图形.

(a) 9x^2 – y^2 = 9

(b) y^2 – 4x^2 = 1

(c) 4x^2 – 9y^2 – 16x – 18y – 29 = 0

Answer (a) :

Answer (b) :

Answer (c) :

(2)

Find the equation of the hyperbola with center at (-7, -2), transverse axis parallel to the x-axis, eccentricity \frac{1}{3}\sqrt{10} , latus rectum\frac{4}{3} .

求中心为(-7, -2), 实轴平行于x轴, 离心率为 \frac{1}{3}\sqrt{10} 及通径长度为 \frac{4}{3} 的双曲线方程式.

Answer :
(x + 7)^2 - 9(y + 2)^2 = 36
(3)

Find the equation of the hyperbola with asymptotes 2x – y = 0, 2x + y – 4 = 0 and passes through the point (6, 10).
求其渐近线为2x – y = 0, 2x + y – 4 = 0且经过点 (6, 10) 的双曲线方程式.

Answer :
(2x – y) (2x + y – 4) = 36
(4)

Find the equation of the hyperbolic with center (3, -6), conjugate axis parallel to the x-axis, distance between foci 6\sqrt5, distance between directrices \frac{24}{5}\sqrt5.

求中心为(3, -6), 虚轴平行于x轴, 焦点之距离为 6\sqrt5 且准线的距离是 \frac{24}{5}\sqrt5 的双曲线方程式

Answer :
(y + 6)^2 - 4(x - 3)^2 = 36
(5)

Given that P is on the right side of the hyperbola\frac{x^2}{9}-\frac{y^2}{16}=1, its distance to the right directrix is \frac{16}{5}, calculate the distance of P to the left focus.

己知双曲线 \frac{x^2}{9}-\frac{y^2}{16}=1 右支上一点P到右准线的距离为 \frac{16}{5}，求P点到左焦点的距离。

Answer :
11\frac{1}{3}
(6)

If the hyperbola \frac{x^2}{9k^2}-\frac{y^2}{4k^2}=1 and circle x^2 + y^2 = 1 don’t have intersection point, find the range of the real value k.

若双曲线 \frac{x^2}{9k^2}-\frac{y^2}{4k^2}=1 与圆 x^2 + y^2 = 1 沒有公共点, 求实数 k 的取值范围.

Answer :
k > \frac{1}{3} or k < -\frac{1}{3}
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