# 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?

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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?