Identifying key points 确定关键点

I’m translating the problem: pinpointing A and B as the ellipse's vertices, computing slopes $k_{AM}$ and $k_{BN}$ for any line through D(1, 0), and establishing their ratio.
我正在翻译这个问题：确定 A 和 B 为椭圆的顶点，计算通过 D(1, 0) 的任意直线的斜率$kAMk_{AM}$kAM和$kBNk_{BN}$kBN，并确定它们的比值。

Mapping the ellipse 绘制椭圆

I’m curious about the ellipse centered at (0,0) with a 2-unit semi-major axis along the x-axis and a 1-unit semi-minor axis along the y-axis. The vertices are at (±2, 0), and point D is at (1, 0).
我对以 (0,0) 为圆心、沿 x 轴有 2 个单位半长轴、沿 y 轴有 1 个单位半短轴的椭圆很好奇。顶点位于 (±2, 0)，D 点位于 (1, 0)。

Parametrizing the line 对直线进行参数化

I’m starting by parametrizing the line through $D(1,0)$ using $y = k(x-1)$ where $k$ is any real number. Next, I’ll find points $M$ and $N$ by solving the system with the ellipse equation.
我首先对通过$D(1,0)$D$0$的直线进行参数化，使用$y = k(x-1)$y = k $(x-$ 1 $)$其中$k$ 是任意实数。接下来，我将通过用椭圆方程求解系统，找到点$MM$M 和$NN$N。

Simplifying the equation 简化方程

I’m thinking through how to express the ellipse equation by plugging in $y$, expanding, and combining terms to form a quadratic equation in $x$. This involves eliminating denominators and simplifying the terms.
我正在思考如何通过插入$yy$ y、展开和合并项来表达椭圆方程，从而在$xx$ x 中形成二次方程。这需要消除分母并简化项。

Mapping slopes 绘制斜坡图

I calculated the slopes $k_{AM}$ and $k_{BN}$ using points A and B on the x-axis. Now, I'm expressing $\frac{k_{AM}}{k_{BN}}$ in terms of $x_M$, $x_N$, $y_M$, and $y_N$.
我利用 x 轴上的 A 点和 B 点计算了斜率$kAMk_{AM}$kAM和$kBNk_{BN}$kBN。现在，我想表达的是$\frac{k_{AM}}{k_{BN}}$ 以$xMx_M$ M, $xNx_N$ N 表示、 $y_M$ ，以及$yNy_N$ yN 。

Identifying relationships
确定关系

I’m establishing the relationship between points M and N on the line $y = k(x - 1)$, determining their y-coordinates based on x-coordinates, and simplifying the ratio of $k_{AM}$ to $k_{BN}$ by canceling the 'k' terms.
我正在直线$y = k(x - 1)$ y = k $(x-$ 1 $)$ 上建立点 M 和 N 之间的关系，根据 x 坐标确定它们的 y 坐标，并简化$kAMk_{AM}$kAM与 <span data-dl-uid="64">$k_{BN}$ ，取消 "k "项。

I’m simplifying the ratio to $\frac{x_M - 1}{x_M + 2} \cdot \frac{x_N - 2}{x_N - 1}$. Next, I’ll explore expressing it in terms of $x_M$ and $x_N$ or finding a relationship between them, referencing the quadratic equation $[1 + 4k^2] x^2 - 8k^2 x + [4k^2 - 4]) = 0$.
我将比率简化为$\frac{x_M - 1}{x_M + 2} \cdot \frac{x_N - 2}{x_N - 1}$.接下来，我将探讨用$xMx_M$xM和$xNx_N$xN来表达，或者参照二次方程$[1 + 4k^2] x^2 - 8k^2 x + [4k^2 - 4]) = 0$[4k2- 4])= 0.

Calculating roots properties
计算根的特性

I’m working through the product and sum of the roots for the quadratic equation $C x^2 + D x + E = 0$, resulting in $x_M + x_N = \frac{8k^2}{1 + 4k^2}$ and $x_M x_N = \frac{4k^2 - 4}{1 + 4k^2}$.
我正在计算一元二次方程$x^2 + D x + E = 0$ 2 + $Dx$+ E = 0 的根的积与和，结果是 $x_M + x_N = \frac{8k^2}{1 + 4k^2}</seman$ 和$x_M x_N = \frac{4k^2 - 4}{1+4k^2}。$ 1+4k24k2-411</span> 。

Deriving expressions 派生表达式

I’m working through equations involving $x_M$ and $x_N$ and expanding expressions like $(x_M - 1)(x_N - 2)$ and $(x_M + 2)(x_N - 1)$.
我正在计算涉及$xMx_M$xM和$xNx_N$xN的方程，并扩展出如下表达式$(x_M - 1)(x_N - 2)$ 和$(x_M + 2)(x_N - 1)$ 2)(xN- 1).

Breaking down the ratio 分解比率

Hmm, I’m thinking through the steps to find the ratio $R$. I’m focusing on the numerator and denominator, which are expressed in terms of $S = x_M + x_N$ and $P = x_M x_N$.
嗯，我正在思考如何找到比率$RR$ R 的步骤。我的重点是分子和分母，它们用$= x_M + x_N$S = x M + x N和$P = x_M x_N$ xMxN .

Breaking down variables 分解变量

OK, let's tally up N and D using S and P, focusing on S and P to simplify calculations. Progressing methodically through each substitution and simplification step-by-step to hone in on a more refined solution.
好，让我们用 S 和 P 来计算 N 和 D，重点是 S 和 P，以简化计算。有条不紊地逐步进行每一次替换和简化，以磨练出更精细的解决方案。

Breaking down the equation
分解等式

I’m simplifying N - D by merging like terms and regrouping, ultimately reducing it to -S - 2 x_N + 4.
我通过合并同类项和重新组合来简化 N - D，最终将其简化为 -S - 2 x_N + 4。

Investigating specific values
调查具体价值

I’m examining unique line values, like $k = 0$ and $k \rightarrow \infty$, to pinpoint intersections with the ellipse, which may partly determine $N / D$.
我正在研究唯一的线值，如$=$k = 0 和 $\rightarrow \infty$ k$\to$ ∞，以确定与椭圆的交点，这可能会部分决定 $/ D$/D。

Determining intersections and slopes
确定交叉口和坡度

Interestingly enough, I’m thinking through intersections and slopes by substituting $x = 1$ into the ellipse equation, finding that the line intersects at points (1, ±$\frac{\sqrt{3}}{2}$). Calculating slopes $k_{AM}$ and $k_{BN}$ and their ratio $\frac{1}{3}$.
有趣的是，我在思考交点和斜率时，将$\mathrm{x=1x}$=$1$x = 1 代入椭圆方程，发现直线交于点（1，±3）。$\frac\{sqrt{3}}{2}$ ）。计算斜率$kAMk_{AM}$kAM和$kBNk_{BN}$kBN及其比率$\frac{1}{3}$ .

Tracing line intersections
追踪直线交叉点

Mapping out where y = x - 1 intersects the ellipse, revealing points M(0, -1) and N($\frac{8}{5}$, $\frac{3}{5}$). Calculating slopes k_{AM} and k_{BN} highlights their ratio.
绘制出 y = x - 1 与椭圆相交的位置，显示出点 M(0，-1) 和 N$85\frac{8}{5}$8 、 $\frac{3}{5}$ ）。计算斜率 k_{AM} 和 k_{BN} 可以突出它们的比率。