I assume that $f(x,y,z)$ is a 2nd degree polynomial whose zero set intersect the plane $\ell x+my+nz=p$ in an ellipse, and that the point $V=(\alpha,\beta,\gamma)$ does not lie in the plane.
Conceptually, we just need to make some coordinate changes which simplify the picture. Let the new coordinates be denoted by $(u,v,w)$. We want the origin in the new coordinates to be the center of the ellipse mentioned above, the plane should be given by the equation $w=0$, the ellipse should be given by the equations $u^2+v^2=1$, $w=0$, and the point $V$ should have new coordinates $(u,v,w)=(0,0,1)$. Then the cone will be given by the equation
Now substitute back to $(x,y,z)$-coordinates.
To carry out this program, find the center $C$ of the ellipse. Let $u$ be the vector from $C$ along the major semiaxis to one of the two points on the ellipse furthest away from $C$. Similarly, let $v$ be the vector from $C$ along the minor semiaxis to one of the two points on the ellipse closest to $C$. Let $w$ be the vector from $C$ to $V$. There are your new coordinates, and the rest is a matter of computation.