Can the pole of a polar line with respect to a non-degenerate conic be inside the polar?

Question

Let $\mathbb{P}$ be a projective space. Consider $Q$ a non-degenerate conic inside $\mathbb{P}$. Let $p \in \mathbb{P}$ be an arbitrary point and let $H_{Q}(p)$ be it's polar line with respect to $Q$. Could it happen that $p \in H_{Q}(p)$?

Answer 1

The answer is "Yes", if and only this point, the pole, is situated onto the conic curve ; in this case, its polar will be the associated tangent line.

Here is an explanation with matrices: dealing with conic curve

$$Ax^2+2Bxy+Cy^2+2Dx+2Ey+F=0 \iff \begin{pmatrix}x&y&1\end{pmatrix}\begin{pmatrix}A&B&D\\B&C&E\\D&E&F\end{pmatrix}\begin{pmatrix}x\\y\\1\end{pmatrix}=0,$$

the equation of the polar of point/pole $(x_0,y_0)$ is given by

$$\begin{pmatrix}x_0&y_0&1\end{pmatrix}\begin{pmatrix}A&B&D\\B&C&E\\D&E&F\end{pmatrix}\begin{pmatrix}x\\y\\1\end{pmatrix}=0 \tag{1}$$

But it is well known that in the particular case where $(x_0,y_0)$ belong to the conic curve, '1) is also the equation of the tangent at this point.

The fact to write :

$$\begin{pmatrix}x_0&y_0&1\end{pmatrix}\begin{pmatrix}A&B&D\\B&C&E\\D&E&F\end{pmatrix}\begin{pmatrix}x_0\\y\\1\end{pmatrix}=0$$

means at the same time that the point belongs to the conic curve AND it belongs to the associated polar line, which is in fact the tangent to the conic curve precisely at this point.