Space geometry

2010016107

Level: 
C
Identify the true statement about the line p:x=t,y=t,z=−2t, t∈R and the sphere κ:(x−3)2+y2+(z−4)2=25.
Line p and sphere κ do intersect in two points.
We do not have enough information to determine whether line p intersects sphere κ.
Line p and sphere κ do intersect in exactly one point.
Line p and sphere κ do not intersect at all.

2010016108

Level: 
C
Identify the true statement about the line q:x=4t,y=t,z=−3t, t∈R and the sphere κ:x2+y2+z2−6x−8z=0.
Line q and sphere κ do intersect in exactly one point.
Line q and sphere κ do not intersect at all.
We do not have enough information to determine whether line q intersects sphere κ.
Line q and sphere κ do intersect in two points.

2010016109

Level: 
C
Identify the true statement about the plane ρ:x+y−z+1=0 and the sphere κ:x2+y2+z2−2x+4y−6z+11=0.
Plane ρ is the tangent plane to sphere κ.
Plane ρ intersects sphere κ and passes through sphere’s center.
Plane ρ and sphere κ have no intersection at all.
Plane ρ intersects sphere κ but does not pass through sphere’s center.

2010016110

Level: 
C
Identify the true statement about the plane σ:2x+y−2z+13=0 and the sphere κ:x2+y2+z2−2x−2y−4z+2=0.
Plane σ and sphere κ have no intersection at all.
Plane σ intersects sphere κ but does not pass through sphere’s center.
Plane σ is the tangent plane to sphere κ.
Plane σ intersects sphere κ and passes through sphere’s center.

2010016112

Level: 
C
Given the sphere (x+1)2+(y+2)2+(z−1)2=4 and the plane 2x−2y+z+d=0, find the parameter d such that the given sphere and the given plane have no intersection at all.
d∈(−∞;−9)∪(3;∞)
d∈(−∞;−3)∪(9;∞)
d∈(−∞;−15)∪(9;∞)
d∈(−∞;−9)∪(15;∞)

2010016113

Level: 
C
Let a point A be the intersection point of the sphere x2+y2+z2−4x−2y+4z−5=0 and z-axis. Find the equations of all the tangent planes to the given sphere at the point A.
2x+y+3z+15=0, 2x+y−3z+3=0
2x+y−3z−15=0, 2x+y+3z−3=0
2x+y+3z+15=0, 2x+y+3z−3=0
2x+y−3z−15=0, 2x+y−3z+3=0

2010016114

Level: 
C
Let a point B be the intersection point of the sphere x2+y2+z2+4x+2y−4z−8=0 and y-axis. Find the equations of all the tangent planes to the given sphere at the point B.
2x−3y−2z−12=0, 2x+3y−2z−6=0
2x+3y−2z+12=0, 2x−3y−2z+6=0
2x−3y−2z−12=0, 2x−3y−2z+6=0
2x+3y−2z+12=0, 2x+3y−2z−6=0