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\thetasin\phi
y=rsin\thetasin\phi
z=rcos\phi


r=1

The Attempt at a Solution

I substitute x,y,z in both equations
d=sin\thetasin\phi+cos\phi+cos\thetasin\phi
cos²\thetasin²\phi+sin²\thetasin²\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.