Not understanding why z is being eliminated.

Question

I am trying to understand a problem from a book, which is given below,

Find the equation of the quadric cylinder with generators parallel to $z$-axis and passing through the curve

$$ax^2+by^2+cz^2=1,\quad lx+my+nz=p.$$

Now, in solution, they have tried to eliminate $z$ in this way.

Here is the solution part (according to the book):

Eliminating $z$ between $ax^2+by^2+cz^2=1$ and $lx+my+nz=p$, we get $$ax^2+by^2+c\biggl(\frac {p-lx-my}{n}\biggr)^2=1.$$

Why are they eliminating $z$? I know that any line parallel to $z$ axis will be represented by $z=0$ equation, but in the above solution provided by the book, they have not used $z=0$. Can anyone please explain what is going on exactly?

Answer 1

To find the desired cyclinder, we need only the $(x,y)$-coordinates of all points of the given curve; their $z$-coordinates do not matter. Eliminating $z$ form the two equations describing the curve in order to arrive at a relation between $x$ and $y$ for points on the curve then seems quite natural.

Answer 2

The answer according to my observation is that when we mention a line parallel to $z\;axis\;$ i guess the $z$ parameter is variable while x and y parameter remains constant no matter how long the line is, thus, $line\;parallel\;to\;z\;axis\;$ makes us sure $x$ and $y$ values are constant but $z$ is not constant, it almost leads to infinity!