Question

Find an equation for the surface consisting of all points P for which the distance from P to the x-axis is 3 times the distance from P to the yz-plane.

Identify the surface.

- hyperboloid of one sheet

Answer 1

The equation for the surface consisting of all points P for which the distance to the x-axis is three times the distance to the yz-plane. In other words, we want an equation such that [
`√(y^2+z^2)`=3 x. Squaring both sides gives us
`y^2+z^2=9x^2`.This is the equation of a hyperboloid of one sheet.

Consider a point

P(x, y, z) in three-dimensional space. The distance from

P to the x-axis is given by
`√(y^2+z^2)`​, and the distance from P to the yz-plane is ∣x∣.

We aim to find an equation for the surface where the distance to the x-axis is three times the distance to the yz-plane, expressed as
`√(y^2+z^2)` = 3x.

Squaring both sides yields
`y^2+z^2=9x^2` This equation represents a specific type of surface in three-dimensional space known as a hyperboloid of one sheet.

Geometrically, a hyperboloid of one sheet resembles a double cone or an hourglass shape.

The equation
`y^2+z^2=9x^2` describes all points P that satisfy the given condition.

The squared terms indicate that the surface extends in both positive and negative directions along the x-axis, creating a symmetrical structure.

In conclusion, the surface defined by the equation
`y^2+z^2=9x^2` is the solution to the geometric problem posed, providing a clear representation of points in three-dimensional space where the specified distance relationship holds.

Question

Find an equation for the surface consisting of all points P for which the distance from P to the x-axis is 3 times the distance from P to the yz-plane.

Identify the surface.

hyperboloid of one sheet

hyperboloid of two sheets

hyperbolic paraboloid

cone

elliptic cylinder

ellipsoid

elliptic paraboloid

parabolic cylinder