The equations of $2$ double cones having their vertex at the origin $(0,0,0)$ are given by:
$(ax+by+cz)^2=(x^2+y^2+z^2)\cos^2(\theta_1) \hspace{25pt} (1)$
($\theta_1=$semi-apical angle, and $a,b,c$ are the direction cosine of the axis of the cone)
$(a'x+b'y+c'z)^2=(x^2+y^2+z^2) \cos^2(\theta_2) \hspace{25pt} (2)$
($\theta_2=$ semi-apical angle, and $a',b',c'$ are the direction cosine of the axis of the cone)
Suppose, it's given that the 2 cones intersect. So, it is obvious that their intersection will yield a pair of straight lines (Also at a specific case only a line, when just touching each other).
What will be the equation of the lines (or line)?