Analytical space geometry

2010016114

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

2010016113

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

2010016112

Level: 
C
Given the sphere \((x + 1)^2 + (y + 2)^2 + (z - 1)^2 = 4\) and the plane \(2x -2 y +z + d = 0\), find the parameter \(d\) such that the given sphere and the given plane have no intersection at all.
\( d \in (-\infty;-9) \cup (3;\infty)\)
\( d \in (-\infty;-3) \cup (9;\infty)\)
\( d \in (-\infty;-15) \cup (9;\infty)\)
\( d \in (-\infty;-9) \cup (15;\infty)\)

2010016111

Level: 
C
Given the sphere \((x - 1)^2 + (y - 2)^2 + (z + 1)^2 = 9\) and the plane \(2x + y - 2z + d = 0\), find the parameter \(d\) such that the intersection of the given sphere and the given plane is a circle.
\( d \in (-15;3)\)
\( d \in (-3;15)\)
\( d \in (-33;21)\)
\( d \in (-21;33)\)