If for any external point, exactly two tangents can be drawn to an algebraic curve, must the curve be a conic?

Question

Yesterday, my teacher, while proving Poncelet's theorem, seemed to use the fact that if from any external point (external meaning, I assume $f(x,y)>0$ where $f$ is the polynomial of two variables corresponding to the curve), exactly two tangents to the curve can be drawn, then the curve is a conic. Is this true? If so, I've not been able to rigorously prove it. Any help will be appreciated. Maybe some additional conditions will be required, like nonsingularity of the curve. I'm not sure.

Answer 1

$$ x^4 + x^2 y^2 + y^4 = 1 $$ is not a conic. All your condition requires is that the curve bound a strictly convex region.