Sphere-plane position

Statement

Given the sphere with centre $(2;-1;3)$ and radius $2$, establish whether the plane $x+y+z=0$ is tangent, secant or external.

Solution

One compares the radius $R=2$ with the distance of the centre from the plane. With $\vec n=(1;1;1)$, $|\vec n|=\sqrt3$: $d = \frac{|2-1+3|}{\sqrt3} = \frac{4}{\sqrt3}\approx 2{,}31.$ Since $d>R$ ($2{,}31>2$), the plane is external to the sphere.

Links

Topics: Analytic geometry in space
Concepts: Point-plane distance · Cartesian plane · Sphere
Skills: Analytic geometry · Reasoning by cases · Using formulae
Exercise types: Geometric problem