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Is it a conic? (Posted on 2017-04-19) Difficulty: 3 of 5
You've probably seen this shape before. It is several straight lines whose outline seems to be curved. Connect the following pairs of points with segments:

(1,0) and (0,8)
(2,0) and (0,7)
(3,0) and (0,6)
...
(8,0) and (0,1)

The 8 segments seem to form the outline a curve, but what curve? Is it a circle or maybe one branch of a hyperbola? Some other conic? Does it's equation have some other nice form?

No Solution Yet Submitted by Jer    
Rating: 4.0000 (3 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Some Thoughts Pen and acrylics | Comment 2 of 7 |
I used to do these in pen, and then in acrylics, but I never stopped to wonder what the curve is.  So, thanks for asking, Jer.

My answer is different from broll's, and he is usually right, but I will post my answer here anyway.

For a given k, the line is y = ((k-9)/k)x + (9-k).

For a value k + d, the line is y = ((k+d-9)/(k+d))x + (9-k-d)

Setting the two y's equal, and solving for x, gives x = k(k+d)/9.
(Hope I did not make a mistake).

The limit as d goes to 0 is x= k^2/9, so this is the point of tangency.

Substituting in the first equation and solving for y gives y = (9-k)^2/9.

So, the curve is sqrt(x) + sqrt(y) = 3.

It is circle-like, but actually it is a special case of a superellipse.  The name is attributable to scientist Piet Hein.  According to the Wikipedia, this special case is a section of a parabola.

Edited on April 19, 2017, 12:25 pm
  Posted by Steve Herman on 2017-04-19 12:08:49

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