Label the given conic $\mathcal{H}.$ Let $\tau$ be the tangent line to $\mathcal{H}$ through $O$ and let $\omega$ be an arbitrary fixed circle tangent to $\tau$ through $O.$ Point $O$ is center of a homology $\mathcal{U}: \omega \mapsto \mathcal{H}.$ Let $OA,OB$ cut $\omega$ again at $A',B',$ i.e. the homologous points of $A,B$ under homology $\mathcal{U}.$ $\angle A'OB' \equiv \angle AOB=90^{\circ}$ $\Longrightarrow$ Lines $A'B'$ always pass through the fixed center $P$ of $\omega,$ thus $AB$ always passes through the image $P'$ of $P$ under $\mathcal{U}.$ Consequently, chords $AB$ pass through a fixed point on the normal line to $\mathcal{H}$ through $O.$