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A variable sphere of constant radius passes through the point $(0, 0, 1)$ and touches the $x y$ -plane. The locus of the centre of the sphere is

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Important Questions on Three Dimensional Geometry

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Sphere

A spherical ball is kept at the corner of a rectangular room such that the ball touches two (perpendicular) walls and lies on the floor. If a point on the sphere is at distance of

9, 16, 25

from the two walls and the floor, then a possible radius of the sphere is

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Sphere

For

x = (x_{1}, x_{2}, x_{3}) \in ℝ^{3}

, define

|x| = \sqrt{x_{1}^{2} + x_{2}^{2} + x_{3}^{2}}

. For

a = (1, 4, 4)

and

b =

(1, 0, 1)

, the maximum value of

|x|

satisfying

|x - a| = 2 |x - b|

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Sphere

The radius of the circle formed when the sphere

x^{2} + y^{2} + z^{2} = 49

is cut by the plane

2 x + 3 y - z - 5 \sqrt{14} = 0

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Sphere

The shortest distance from the origin to a variable point on the sphere

{(x - 2)}^{2} + {(y - 3)}^{2} + {(z - 6)}^{2} = 1

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Sphere

If two spheres of radii

r_{1}

and

r_{2}

cut orthogonally, then the radius of the common circle is

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Sphere

The smallest radius of the sphere passing through (1, 0, 0), (0, 1, 0) and (0, 0, 1) is

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Sphere

The centre of sphere passes through four points

(0, 0, 0), (0,2, 0), (1,0, 0) a n d (0, 0, 4)

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Sphere

The centre and radius of the sphere

x^{2} + y^{2} + z^{2} + 3 x - 4 z + 1 = 0

are

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Sphere

Position vector of the center of the sphere

{| \vec{r} |}^{2} + \vec{r} . (\hat{i} + \hat{j} + \hat{k}) - 6 = 0

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Sphere

(2, 3, 5)

is one end of a diameter of the sphere

x^{2} + y^{2} + z^{2} - 6 x - 12 y - 2 z + 20 = 0

then co-ordinates of the other end of the diameter are

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Sphere

How many different sphere of radius '

r

' can be drawn which touches all the three co-ordinate axes

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Sphere

The shortest distance from the point $(1, 2, - 1)$ to the surface of the sphere $x^{2} + y^{2} + z^{2} = 24$ is

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Sphere

The equation of the sphere concentric with the sphere

2 x^{2} + 2 y^{2} + 2 z^{2} - 6 x + 2 y - 4 z = 1

and double its radius is

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Sphere

Let

(3, 4, - 1)

and

(- 1, 2,3)

are the end points of a diameter of sphere. Then, the radius of the sphere is equal to

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Sphere

The shortest distance from the plane

12 x + 4 y + 3 z = 327

to the sphere

x^{2} + y^{2} + z^{2} + 4 x - 2 y - 6 z = 155

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Sphere

If the plane

2 a x - 3 a y + 4 a z + 6 = 0

passes through the midpoint of the line joining the centres of the spheres

x^{2} + y^{2} + z^{2} + 6 x - 8 y - 2 z = 13

and

x^{2} + y^{2} + z^{2} - 10 x + 4 y - 2 z = 8

, then

a

equals

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Sphere

The radius of sphere

x + 2 y + 2 z = 15

and

x^{2} + y^{2} + z^{2} - 2 y - 4 z = 11

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Sphere

Co-ordinate of a point equidistant from the points

(0,0, 0), (a, 0, 0), (0, b, 0), (0, 0, c)

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Sphere

The centre of sphere passes through four points

(0, 0, 0)

(0, 2, 0), (1, 0, 0)

and

(0, 0, 4)

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Sphere

The equation of the sphere touching the three co-ordinate planes is

A variable sphere of constant radius passes through the point 0, 0, 1 and touches the xy-plane. The locus of the centre of the sphere is

Important Questions on Three Dimensional Geometry

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A variable sphere of constant radius passes through the point $(0, 0, 1)$ and touches the $x y$ -plane. The locus of the centre of the sphere is