The locus and the ellipse

A reader in Chile shared a great story with me recently.

Raul is 85 years old, and for over 60 years he's been bugged by a question that his professor asked him during an oral examination. I'll let Raul tell his story, which shows cool evidence of Web-savviness. 

In late 1941, in an oral exam, my distinguished Professor Don Raul Valdivieso, raised the following issue:

"Find the locus of the centre of an ellipse that moves tangent to the positive x- and y-axes."

After brief reflection, I remembered a problem posed in Course Geometry Plana (year 1940) by the eminent Professor Don Luciano Claude which read as follows:

"Find the locus of the point where two straight orthogonal lines intersect, and which are tangential to a given ellipse."

The solution to this problem, easy to find in any treaty on conics, is a concentric circle to an ellipse given with the radius  equal to:

√(a² + b²), where a and b are the semi-axis of ellipse.

Explanation: "Orthogonal" means "at right angles" and as we move the pair of orthogonal lines around the ellipse so that they touch the ellipse at one point (they are tangents), a circle is formed by the intersection point:

Here's the resulting circle, which is concentric with the ellipse (this means it has the same centre).

Back to Raul's story:

So it is easy to deduce that the inverse of this problem is to consider the orthogonal straight tangents as fixed and  the ellipse given as mobile, returning to the problem of Professor Valdivieso.

Clearly the locus will be a portion of the arc of a circle.

However, Professor Valdivieso, even accepting that this was a good solution,  required me to follow the long road, namely the analytic geometric deduction directly as initialy requested.

Result: I was not able to find such a solution.

The years passed and my attempts were in vain to find a solution following the method that I had being requested. Several colleagues whom I consulted could not  resolve it.

So, already retired and close to fulfilling 85 years, I went back to this issue. The Google Scholar was no help to find what I was are looking for. Finally, using the Central Library at the Catholic University and recalling those old fragile, but extraordinary French books, I found what I was looking for in Lecons de Géométrie analytique. Author: Charles Roguet. Edition year 1860.  Page 218.

We need to resolve the previously stated locus,to develop:

SECTION (A) the equation of a tangent line to the ellipse and

SECTION (B) The equation of the distance from a given point to a given straight line.

What follows is quite a lengthy analysis using the characteristics of conic sections.

So dear Reader, can you solve it given the hints above?

The final result is the red arc that you see in the following diagram (the yellow and green curves are not relevant to this problem, even though they are interesting):

Image source: MathCurve.

Raul finished his note as follows, with a challenge to find a simpler solution than his 4 pages of algebra:

That was it. To this day I can not explain that Professor Raul Valdivieso expected that any students could develop this problem, in an examination or in any other circumstance.

It there a much simpler solution? I venture to think there has to be one, because the distinguished professor was very demanding but very thoughtful and fair.

Raúl Espinosa Wellmann, Civil Engineer, Pontificia Universidad Católica de Chile.

Thanks, Raul for the interesting story and for the evidence that a well-crafted problem is simple to state, and motivating for years.

 

Update: A reader asked for the full solution, and so here you go. I think you'll agree it is quite an extraordinary question for an examination!

Locus of an ellipse [MS Word document]

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