求一个长轴为 50 个单位，短轴为 20 个单位的水平椭圆方程

圆锥截面也称为圆锥曲线，是在平面与圆锥相交时形成的。这些部分的形状取决于交叉点发生的角度。因此有四种类型的圆锥曲线，即圆、椭圆、抛物线和双曲线。这些形状中的每一个都有自己的属性和数学方程式。下面讨论椭圆。

椭圆

椭圆是一种圆锥截面，当平面以 0 到 90 度之间的角度与圆锥相交时形成。结果，它概括了一个圆，这是一种特殊的椭圆，具有相同的两个焦点。

椭圆的偏心率 e 是一个范围从 e=0 到 e=1 的量，它显示了它沿每个轴的拉长程度。椭圆有两个轴，分别称为长轴和短轴。在水平椭圆的情况下，沿 x 轴的轴称为长轴，沿 y 轴的轴称为短轴。

椭圆方程

以原点为中心的椭圆方程的标准形式如下：

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

where a and b are the semi-major and minor axes respectively. 

需要记住的是，如果 a > b，那么我们手头有一个水平椭圆。否则，如果 a < b，则它是一个垂直椭圆。

求一个长轴为 50 个单位，短轴为 20 个单位的水平椭圆方程。

解决方案：

The standard form of equation of a horizontal ellipse with its center at the origin is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, a > b.

Given: Length of major axis = 2a = 50 units

⇒ a = 50/2 = 25 units

Length of minor axis = 2b = 25 units

⇒ b = 20/2 = 10 units

Substituting the values in the equation, we have:

$\frac{x^2}{25^2}+\frac{y^2}{10^2}=1$

类似问题

问题 1. 为一个长轴为 60 个单位，短轴为 20 个单位的水平椭圆做一个方程？

解决方案：

The standard form of equation of a horizontal ellipse with its center at the origin is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, a > b.

Given: Length of major axis = 2a = 60 units

⇒ a = 60/2 = 30 units

Length of minor axis = 2b = 20 units

⇒ b = 20/2 = 10 units

Substituting the values in the equation, we have:

$\frac{x^2}{30^2}+\frac{y^2}{10^2}=1$

问题 2. 为一个长轴为 30 个单位，短轴为 10 个单位的水平椭圆做一个方程？

解决方案：

The standard form of equation of a horizontal ellipse with its center at the origin is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, a > b.

Given: Length of major axis = 2a = 30 units

⇒ a = 30/2 = 15 units

Length of minor axis = 2b = 10 units

⇒ b = 20/2 = 5 units

Substituting the values in the equation, we have:

$\frac{x^2}{15^2}+\frac{y^2}{5^2}=1$

问题 3. 为一个长轴为 70 个单位，短轴为 40 个单位的水平椭圆做一个方程？

解决方案：

The standard form of equation of a horizontal ellipse with its center at the origin is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, a > b.

Given: Length of major axis = 2a = 70 units

⇒ a = 70/2 = 35 units

Length of minor axis = 2b = 40 units

⇒ b = 40/2 = 20 units

Substituting the values in the equation, we have:

$\frac{x^2}{35^2}+\frac{y^2}{20^2}=1$

问题 4. 为一个长轴为 100 个单位，短轴为 20 个单位的水平椭圆做一个方程？

解决方案：

The standard form of equation of a horizontal ellipse with its center at the origin is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, a > b.

Given: Length of major axis = 2a = 100 units

⇒ a = 100/2 = 50 units

Length of minor axis = 2b = 20 units

⇒ b = 20/2 = 10 units

Substituting the values in the equation, we have:

$\frac{x^2}{50^2}+\frac{y^2}{10^2}=1$

问题 5. 为一个长轴为 42 个单位，短轴为 6 个单位的水平椭圆做一个方程？

解决方案：

The standard form of equation of a horizontal ellipse with its center at the origin is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, a > b.

Given: Length of major axis = 2a = 42 units

⇒ a = 42/2 = 21 units

Length of minor axis = 2b = 6 units

⇒ b = 6/2 = 3 units

Substituting the values in the equation, we have:

$\frac{x^2}{21^2}+\frac{y^2}{3^2}=1$