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第4章:坐标变换
Lesson: 5.4 Hyperbola 双曲线
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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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