Vector equation for the tangent line of the intersection of $x^2 + y^2 = 25$ and $y^2 + z^2 = 20$ What is the vector equation for the tangent line of the intersection of $x^2 + y^2 = 25$ and $y^2 + z^2 = 20$ at the point $(3,4,2)$?
I think I should find a vector
$$
\gamma(t) = (x(t),y(t),z(t))
$$
that represents the intersection. Because then, I can find easily the tangent line.
But I do not know a general technique to find such a $\gamma$.
Any suggestion please?

Hint: Maybe use $y=t,\ x=(25-t^2)^{1/2},\ z=(20-t^2)^{1/2}$ and restrict $t$ to make the radicals defined. I think the tangent line would then be possible except at the endpoints, for specific values of $t$ chosen.
ADDED: Actually $x$ and/or $z$ could be chosen as the negatives of the above radicals, and that would also be part of the intersection. So one has to choose specific signs for these at each given point. (This means for a fixed $t_0$ to choose either positive or negative radicals for the $x,z$ coordinates to get both the point on the intersection and the value of the derivative vector at that point.)
