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第4章:坐标变换
高三数学 | 高级数学
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5.3 Ellipse 椭圆


1. Find the centre, vertex, axis of symmetry, major semi-axis, minor semi-axis, eccentricity, focus, directrix and length of latus rectum for the following ellipse and sketch the graph.
求以下各椭圆的中心, 顶点, 对称轴, 半长轴之长, 半短轴之长, 离心率, 焦点, 准线和通径长度並描其图形

(a) \frac{x^2}{50}+\frac{y^2}{32}=2

(b) 25x^2 + 9y^2 = 225

(c) 4x^2 + y^2 + 16x- 10y- 23 = 0

(d) 4x^2+ 9y^2 - 16x + 18y - 11 = 0

(e) 25x^2 + 16y^2 -150x + 32y - 159 = 0

Answer (a) :

Answer (b) :

Answer (c) :

Answer (d) :

Answer (e) :

2. Find the equation of the ellipse with center at (-3, 1), one of vertices at (-5, 1), length of latus rectum equal to 1 and
求椭圆方程式其中心在(-3, 1), 其中一顶点在(-5, 1), 通径长度为1且

(a) major axis parallel x-axis,
长轴平行于x轴,

(b) major axis parallel y-axis,
长轴平行于y轴,

Answer :
(a) (x + 3)^2 + 4(y - 1)^2 = 4

(b) \frac{{(x+3)}^2}{4}+\frac{{(y-1)}^2}{64}=1

3. Find the equation of the ellipse which has same eccentricity and left directrix with ellipse x^2 + 2y^2 = 2, and also take its right focus to be the left focus.

求与椭圆x^2 + 2y^2 = 2有相同的离心率和公共的左准线,且以它的右焦点为左焦点的椭圆方程.

Answer :
\frac{{(x-4)}^2}{18}+\frac{y^2}{9}=1
4. Given that the major axis of an ellipse is 4, y-axis be its directrix, the left focus is on the parabola y^2 = x - 1. Find the equation of the ellipse when eccentricity is \frac{2}{3}.

己知椭圆的长轴长为4,以y轴为准线,左焦点在拋物线y^2 = x - 1上。求当离心率为\frac{2}{3}时, 此椭圆方程式。

Answer :
\frac{{(x-3)}^2}{4}+\frac{9{(y\pm\sqrt{2/3})}^2}{20}=100
5. Given that F_1(2, -2), F_2(2, 0) are the foci of a ellipse, straight line y = 3 is a directrix of the ellipse, find the equation of the ellipse.

已知椭圆的焦点为F_1(2, -2), F_2(2, 0), 直线y = 3是椭圆的一条准线, 求椭圆的方程.

Answer :
\frac{{(x-2)}^2}{3}+\frac{{(y+1)}^2}{4}=1
6. Find the equation of the ellipse with vertices at (0, -1) and (12, -1), a focus at (6 + \sqrt{11}, -1).

求顶点在 (0, -1) 和 (12, -1), 一焦点在 (6 + \sqrt{11}, -1) 的椭圆方程式

Answer :
\frac{{(x-6)}^2}{36}+\frac{{(y+1)}^2}{25}=1
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