Help with finding an equation!

[Pokerman2627]

Hey guys! Im stuck on this question, can anyone help me out?

Find the equation for the surface consisting of all points P whose distance from the x-axis is twice the distance from the YZ plane. What is the name of this surface?

 

[tkhunny]

If a point in 3-space is (a,b,c)...

You'll kick yourself... "Distance from the YZ Plane?" Isn't that "a", the x-coordinate? Don't let the easy ones get away from you.

You do the slightly harder one.

 

[HallsofIvy]

I would write a point in R³ as (x, y, z). "The distance from a point to a set of points" always means the shortest distance from the given point to a point in the given set. And, by the Pythagorean theorem, that is along a line perpendicular to the set.

So the "distance from the x-axis" is the distance from (x, y, z) to (x, 0, 0). What is the distance between those two points? And the "distance to the yz-plane" is the distance from (x, y, z) to (0, y, z). What is the distanced between those two points?

 

[pka]

Find the equation for the surface consisting of all points P whose distance from the x-axis is twice the distance from the YZ plane. What is the name of this surface?

Notation: We will use \(\Pi_{YZ}\) for the \(YZ\)-plane and \(\ell_x\) for the \(x\)-axis.
Now for the distance formulas. If \(P: (a,b,c)\) then:
1) \(d(P,\Pi_{YZ})=|a|\)
2) \(d(P,\ell_x)=\sqrt{b^2+c^2}\).