For a point mass, $\tau=I\alpha$ and $L=I \omega$.
What if we apply a force to the edge of an ellipse, if the torque is $rF\sin\theta$, where $r$ is the distance between the point where the force is applied to the centre of mass of the ellipse, then the equation relating angular momentum and this force will then be $rFt\sin\theta=I\omega$ or $mvr\sin\theta=I\omega$, where "I" will be the moment of inertia of ellipse about its centre of mass. However, $mvr\sin\theta$ is the angular momentum of a point mass instead of an ellipse.
What is the mistake? Thank you.