# A conic on three points and a pole-polar pair

Please answer this a question , how to find an conic which passes 3 given points and has the given pole-polar pair. Does this determine the conic uniquely?

-

## 2 Answers

The question is not very clear, so I have to make some assumptions. Let's give the names $A$, $B$, $C$ to the three points that the conic passes through (in that order).

When you say "has the given pole-polar pair", I'm assuming you mean that we are given some fourth point $D$ which is the pole corresponding to the line $AB$.

If my assumptions are correct, then this data does indeed determine the conic uniquely. In general, you need five point/tangent conditions to determine a conic. We have three points, obviously, and the fourth point $D$ gives you tangent lines $DA$ and $DC$.

When you say "find" the conic, what is it that you want to know about it? Or, saying it another way, in what form would you like to "know" it?

Whatever you want, this page probably has the answer.

-
"The question is not very clear", hence explanation: Given: four points A,B,C and a line p and a point P. Construct (with the help of an unmarked ruler only) a conic that passes through A,B,C and such that the line p is the polar of the point P. It is not certain that these five conditions determine the conic uniquely. For example, there are in general two different conic on four given points and tangent to one additional line. – PeterPSP Feb 26 '13 at 19:20
Please take a look at the web page I referenced in the last line of my answer (the "this page" reference). It shows how three points and two tangents determine a conic. – bubba Feb 27 '13 at 6:38

If I remeber correctly I think that you must first find the correct coordinates where this comic point is located. Then find that the the thrid dimensional point on the grid as well.

-