My questions concerns a chapter in Michael Spivak's math book Calculus. On page 81, Spivak talks about finding a coordinate axes for a plane P that intersects a cone. What I don't understand is the part where he talks about a point in that plane having a certain first coordinate with respect to the newly created axes; he labels this as x. This same point has αx+β its first coordinate relative to the standard axes with which we are familiar. I don't understand where he got the αx+β. I really hope someone clears this up for me.
Here is a quote from that page dealing with conic sections: "Now we have to choose coordinate axes in the plane P. We can choose L as the first axis, measuring distances from the intersection Q with the horizontal plane (Figure 5); for the second axis we just choose the line through Q parallel to our original second axis. If the first coordinate of a point in P with respect to these axes is x, then the first coordinate of this point with respect to the original axes can be written in the form αx+β for some β. On the other hand, if the second coordinate of the point with respect to these axes is y, then y is also the second coordinate with respect to the original axes."