# Range of the parameter of sphere intersecting with a plane

1. Oct 19, 2009

1. The problem statement, all variables and given/known data
Find the range of the parameter d for which the intersection of the sphere x2+y2+z2=1 and the plane x+y+z=d is non-empty.

2. Relevant equations
Cartesian coordinates of a sphere:
x=rcos$$\theta$$sin$$\phi$$
y=rsin$$\theta$$sin$$\phi$$
z=rcos$$\phi$$

r=1

3. The attempt at a solution
I substitute x,y,z in both equations
d=sin$$\theta$$sin$$\phi$$+cos$$\phi$$+cos$$\theta$$sin$$\phi$$
cos2$$\theta$$sin2$$\phi$$+sin2$$\theta$$sin2$$\phi$$+cos2$$\phi$$=1

Since sin2$$\theta$$+cos2$$\theta$$=1
I get 1+cos2$$\phi$$=1
This implies that $$\phi$$=90
Which solves the first equation for d=sin$$\theta$$+cos$$\theta$$
Is this right?
Thanks for any help.

2. Oct 19, 2009

### LCKurtz

I would think your answer would need to be in the form a ≤ d ≤ b. A simpler approach might be to observe the plane will intersect the sphere if the distance of the plane from the origin is ≤ 1. This can be easily done with vectors and no need for spherical coordinates.