Condition for a plane to intersect a sphere

Question

Write down the condition to be satisfied by the numbers $a, b, \ \& \ c$ in order that the sphere $x^2+y^2+z^2=1$ and the plane $ax+by+cz+d=0$ have a non-empty intersection.

I am currently working through a set of general math problems. I cant seem to get through these geometry problems. How do I do this one?

Answer 1

The distance from the plane $\pi :ax+by+cz+d=0 $ to the center $O(0,0,0)$ of the sphere is $$d=d(0,\pi)=\left|\frac{d}{\sqrt{a^2+b^2+c^2}}\right|$$ There is non empty intersecction iff $d\le 1$ (ratius of the sphere). Equivalently, iff $$d^2\le a^2+b^2+c^2.$$