In the previous article, "A Fixed-Length Chord in a Parabola", we solved the problem of the locus of the midpoint of a fixed-length chord within a parabola, which was a relatively simple problem. Although both are conic sections, extending the same problem to an ellipse is not quite as straightforward. This is because the x and y coordinates in the trajectory equation of an ellipse are "entangled" through their squares, unlike the parabola equation which can be easily separated (referring to separation into the form y=f(x)). BoJone tried several methods but found it difficult to derive the trajectory. Finally, through the "Circle Transformation Method", the trajectory equation was eventually obtained.
The so-called Circle Transformation Method involves stretching the ellipse into a circle and utilizing the properties of a circle to solve problems. As is well known, compared to an ellipse, a circle possesses considerably more simplicity. This is a method I summarized while studying various college entrance examination conic section problems before my own exams. Sometimes, after stretching an ellipse into a circle, the conclusion becomes quite obvious; meanwhile, since a circle is a special case of an ellipse, general conclusions for ellipses naturally hold for circles as well. Therefore, to study ellipse problems, one might as well first study the special case—the circle; on the other hand, utilizing the symmetry of the circle can significantly reduce the amount of calculation, which is why BoJone likes this method so much. Even more unexpectedly, it actually came in handy for finding the trajectory in this article.
How do we transform an ellipse into a circle? It is very simple. Suppose the equation of the ellipse is \frac{x^2}{a^2}+\frac{y^2}{b^2}=1. We only need to stretch the y-axis coordinates to \frac{a}{b} times their original value, and the ellipse becomes x^2+y^2=a^2; or contract the x-axis coordinates to \frac{b}{a} times their original value, and the ellipse becomes x^2+y^2=b^2.
What are the characteristics of the figure after stretching or shrinking? The original relationships of intersection, tangency, and separation remain unchanged. The area of the figure changes according to the ratio of the length extension (since only one dimension is stretched, it is not squared). Of course, angles and lengths will change, as the reader will experience in the following calculations.
As shown in the figure, there is a fixed-length chord AB in the ellipse \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, where |AB|=2l. P(x,y) is the midpoint of AB, and let the slope of AB be k.
Using the stretching method, we stretch the y-axis coordinates of the ellipse to \frac{a}{b} times the original, yielding the following results:
Point P becomes P'(x',y')=P'(x,\frac{a}{b}y).
The slope k becomes k'=\frac{a}{b}k.How does the length change? We have the chord length formula |AB|=\sqrt{1+k^2}|x_A-x_B|. It can be seen that the chord length is proportional to \sqrt{1+k^2}, so the length of the chord after stretching is: |A'B'|=2l\cdot \sqrt{\frac{1+\frac{a^2}{b^2}k^2}{1+k^2}}
It is easy to obtain the slope of OP': \frac{y'}{x'}=\frac{ay}{bx}=-\frac{1}{k'}=-\frac{b}{ak} (since in a circle, the radius to the midpoint of a chord is perpendicular to the chord).
From this, we solve for k: k=-\frac{b^2 x}{a^2 y}.
We can also calculate the length of OP': |OP'|^2=x'^2+y'^2=x^2+\frac{a^2}{b^2}y^2
On the other hand, it is also equal to: a^2-\left(\frac{|A'B'|}{2}\right)^2=a^2-l^2\left(\frac{1+\frac{a^2}{b^2}k^2}{1+k^2}\right)
That is: x^2+\frac{a^2}{b^2}y^2=a^2-l^2\left(\frac{1+\frac{a^2}{b^2}k^2}{1+k^2}\right)
Substituting the derived k=-\frac{b^2 x}{a^2 y} into the above equation and performing a series of simplifications, we obtain: a^2 b^2\left[1-l^2\left(\frac{a^2 y^2+b^2 x^2}{a^4 y^2+b^4 x^2}\right)\right]=a^2 y^2+b^2 x^2
This is the trajectory equation! Of course, it can also be simplified into the following result. In my opinion, the following formula looks better: \sqrt{a^2 b^2-a^2 y^2-b^2 x^2}\cdot \sqrt{\frac{a^4 y^2+b^4 x^2}{a^2 y^2+b^2 x^2}}=lab (This result was obtained on 2012.06.05)
This is the result when a=2, b=1, l=0.5:
Summary
The "Circle Transformation Method" is a geometric transformation method and an effective way to handle the heavy calculations found in conic section exercises. Of course, it has its limitations. In particular, this method often fails when encountering fixed-angle problems; specific situations require the reader to explore and summarize more in practical applications. In the process of high school learning, teachers almost never teach this technique for the sake of uniformity. However, I like it deeply because of its versatility and conciseness. Although it has a bit of a "folk remedy" flavor, it truly has many applications. Interested friends might as well think more from this perspective when encountering related problems in the future; perhaps there will be unexpected gains. On the other hand, many geometric conclusions that are quite simple in a circle (such as the Secant-Tangent Theorem) become excellent conic section exercises when transformed into an ellipse through stretching or shrinking. This may also be a line of thought for those who design conic section problems.
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