Finding line through a cone

Question

Question: Find a parameterization for the line that passes through the point $(8,-1,1)$ and contained in the cone:

$$(\frac{x-3}{5})^2+(\frac{y+1}{5})^2=(\frac{z+1}{2})^2$$

Here is what I tried:

If the line is contained inside the cone it should pass through the tip. The tip should be $(3,-1,-1)$

We have two points so one can find the line to be:

$L:(8,-1,1)+t(3,-1,1)$

However, I get a false answer for this answer.

Where am I wrong?

Answer 1

An equation of the line from a point $P$ to a point $Q$ is $r(t) = (1-t)P + tQ, t\in [0,1]$.