Analytical Space Geometry

2010016105

Level: 
C
Let \(C = [2, -4, 3]\) and \(D = [-1, -1, 9]\). Find the intersection points of the sphere \((x − 1)^2 + (y + 3)^2 + (z - 2)^2 = 9\) and the ray that is opposite to the ray \(CD\).
\( [3,-5,1]\)
\( [3,-5,1]\), \( [1,-3,5]\)
\( [-3,5,-1]\), \( [-1,3,-5]\)
\( [1,-3,5]\)

2010016107

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

2010016108

Level: 
C
Identify the true statement about the line \(q: x = 4t, y = t, z = -3t\), \(t \in \mathbb{R}\) and the sphere \(\kappa : x^2 + y^2 + z^2-6x-8z = 0\).
Line \(q\) and sphere \(\kappa\) do intersect in exactly one point.
Line \(q\) and sphere \(\kappa\) do not intersect at all.
We do not have enough information to determine whether line \(q\) intersects sphere \(\kappa\).
Line \(q\) and sphere \(\kappa\) do intersect in two points.

2010016109

Level: 
C
Identify the true statement about the plane \(\rho : x + y - z + 1 = 0\) and the sphere \(\kappa : x^2 + y^2 + z^2 - 2x + 4y - 6z + 11 = 0\).
Plane \(\rho\) is the tangent plane to sphere \(\kappa\).
Plane \(\rho\) intersects sphere \(\kappa\) and passes through sphere’s center.
Plane \(\rho\) and sphere \(\kappa\) have no intersection at all.
Plane \(\rho\) intersects sphere \(\kappa\) but does not pass through sphere’s center.

2010016110

Level: 
C
Identify the true statement about the plane \(\sigma : 2x + y - 2z + 13 = 0\) and the sphere \(\kappa : x^2 + y^2 + z^2 - 2x -2y - 4z + 2 = 0\).
Plane \(\sigma\) and sphere \(\kappa\) have no intersection at all.
Plane \(\sigma\) intersects sphere \(\kappa\) but does not pass through sphere’s center.
Plane \(\sigma\) is the tangent plane to sphere \(\kappa\).
Plane \(\sigma\) intersects sphere \(\kappa\) and passes through sphere’s center.

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.
\( d \in (-15,3)\)
\( d \in (-3,15)\)
\( d \in (-33,21)\)
\( d \in (-21,33)\)

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.
\( d \in (-\infty,-9) \cup (3,\infty)\)
\( d \in (-\infty,-3) \cup (9,\infty)\)
\( d \in (-\infty,-15) \cup (9,\infty)\)
\( d \in (-\infty,-9) \cup (15,\infty)\)

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 \(A\).
\(2x + y + 3z + 15 = 0\), \(2x + y - 3z + 3 = 0\)
\(2x + y - 3z -15 = 0\), \(2x + y + 3z - 3 = 0\)
\(2x + y + 3z + 15 = 0\), \(2x + y + 3z - 3 = 0\)
\(2x + y - 3z - 15 = 0\), \(2x + y - 3z + 3 = 0\)

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 \(B\).
\(2x -3y -2z -12 = 0\), \(2x + 3y - 2z -6 = 0\)
\(2x + 3y - 2z +12 = 0\), \(2x -3 y -2z +6 = 0\)
\(2x -3y -2z -12 = 0\), \(2x -3 y -2z +6 = 0\)
\(2x + 3y - 2z +12 = 0\), \(2x + 3y - 2z -6 = 0\)