I want to solve this question:
Find the equation for the set of points equidistant from the $y$-axis and the plane $z=6$.
My attempt:
The $y$-axis has the equation $x=0$. The distance from any point to the plane $z=6$ is the absolute value of that point's z-coordinate minus $6$ or $|z-6|$.
Is the equation $|x-0|=|z-6|$? And is the distance from any point to the $y$-axis $|x'-0|$ where $x'$ is the $x$ coordinate of that arbitrary point?
Even If I'm correct, I really don't understand why or have an intuitive sense on how to find points equidistant from axis, planes, lines, etc. Can I get help on how to develop intuition for knowing equidistance?
I think it's $$\sqrt{x^2+z^2}=|z-6|$$