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Byju's Answer
Standard XII
Mathematics
Equally Likely Events
we are to tak...
Question
we are to take 4 points in the surface of a sphere. what is the (average)probability that the tetrahedron formed by the 4 points will contain the centre of the sphere in it
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Q.
A tetrahedron is a three dimensional figure bounded by four non-coplanar triangular planes. So, a tetrahedron has four non-coplanar points as its vertices.
Let a tetrahedron has four points A, B, C, D whose co-ordinates are
(
x
i
,
y
i
,
z
i
)
, i=1, 2, 3, 4 respectively in a rectangular three dimensional space. The co-ordinates of its centroid are given by
∑
4
i
=
1
x
i
4
,
∑
4
i
=
1
y
i
4
,
∑
4
i
=
1
z
i
4
. The circumcenter of the tetrahedron is the center of a sphere passing through its vertices. So this point is equidistant from each of vertices of tetrahedron.
Let tetrahedron has three of its vertices represented by the points A (6, - 5, -1), B(- 4, 1, 3) & C(2, -4, 18) & its centroid lies at the point (1, -2, 5).
On the basis of above information answer the following questions.
The co-ordinate of centre of the sphere circumscribe the tetrahedron is
Q.
Consider a sphere passing through the origin and the points
(
2
,
1
,
−
1
)
,
(
1
,
5
,
−
4
)
,
(
−
2
,
4
,
−
6
)
.
What is the centre of the sphere?
Q.
A metal sphere has radius
R
and mass
M
. A spherical hollow of diameter
R
is made in this sphere such that its surface passes through the centre of the metal sphere and touches the outside surface of the metal sphere. A unit mass is placed at a distance from the centre of metal sphere. The gravitational field at that point is
Q.
A sphere is pure rolling on a horizontal surface such that its centre moves with 5 m/s as shown in figure. What is the magnitude of velocity of point B on this sphere?
Q.
A sphere is rolling without slipping on a fixed horizontal plane surface. In the figure,
A
is the point of contact,
B
is the centre of the sphere and
C
is its topmost point. Then,
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