Range of the parameter of sphere intersecting with a plane

[Jadehaan]

Homework Statement

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.

Homework Equations

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


r=1

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.

 

[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.