Three Dimensional Geometry Test

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Three Dimensional Geometry Test - 3

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Weekly Quiz Competition

Question 1
1 / -0

The locus of a point which moves so that the difference of the squares of its distances from two given points is constant, is a:

A
straight line

B
plane

C
sphere

D
none of these
Question 2
1 / -0

The radius of the sphere through the points ( 4 ,3 , 0 ) , ( 0 , 4 , 3 ) ,( 0 , 5 , 0 ) and ( 4 , 0 , 3 ) is

A
7

B
7/ 5

C
5/ 7

D
5

Solution

we know that general equation of sphere is x2+y2+z2+2gx+2fy+2hz+c=0

since sphere passing through the points ( 4 ,3 , 0 ) , ( 0 , 4 , 3 ) ,( 0 , 5 , 0 ) and ( 4 , 0 , 3 ) so putting these values one by one in given equation , we get

16+ 9+ 8g + 6f + c=0 =>8g +6f +c = -25 -------------1)

16 +9 +8f +6h +c=0 =>8f + 6h +c= -25 -----------2)

25 + 10f +c = 0 -----------------------3)

16+ 9+ 8g + 6h +c=0 =>8g +6h +c= -25 -------------4)

by 2) - 4 ) we have f = g

using this in 1) we have 16f +c = -25 -------5)

Now solving 3) and 5) we get f=0

using value of f we have g= 0 , h= 0 , c= -25

Now radius =√(−g)2+(−f)+(−h)2−c = √25 =5
Question 3
1 / -0

The direction ratios of normal to the plane through (1, 0, 0) and (0, 1, 0), which makes an angle π/4 with the plane x + y = 3, are

A
1, √2, 1

B
1, 1, √2

C
1, 1, 2

D
√2, 1, 1
Question 4
1 / -0

If the direction cosines of a straight line are < k , k , k > , then what will be the value of k

A
k > 0

B
k = 1

C
0 < k < 1

D
k=1√3 or k=-1√3

Solution

We know that if l , m , n are direction cosines of a lines then l² +m²+n²= 1

so, k² + k² +k² = 1

3 k² = 1

k²= 1/3

taking squareroot on both sides , we have k=1/√3ork=−1/√3
Question 5
1 / -0

The angle between the lines x = 1, y = 2 and y = 1, z = 0 is

A
90^o

B
30^o

C
60^o

D
0^o
Question 6
1 / -0

the numbers 3, 4 , 5 can be

A
coordinates of a point on the line y = 4 , z = 0

B
coordinates of a point in the plane x + y – z = 0

C
direction numbers of a line in space

D
direction cosines of a line in space

Solution

the numbers 3, 4 , 5 can be direction ratio of any line these not satisfying any other option
Question 7
1 / -0

The cosine of the angle between any two diagonals of a cube

A
1/3

B
1/2

C
2/3

D
1/√3
Question 8
1 / -0

The shortest distance from the plane 12x + 4y + 3z = 327 to the sphere x² + y² + z² + 4x – 2y – 6z = 155 is:

A
26

B
11 4/3

C
13

D
39

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Re-Attempt Weekly Quiz Competition

Three Dimensional Geometry Test - 3

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Three Dimensional Geometry Test - 3

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SHARING IS CARING

Question 1

straight line

plane

sphere

none of these

Question 2

7

7/ 5

5/ 7

5

Solution

Question 3

1, √2, 1

1, 1, √2

1, 1, 2

√2, 1, 1

Question 4

k > 0

k = 1

0 < k < 1

k=1√3 or k=-1√3

Solution

Question 5

90o

30o

60o

0o

Question 6

coordinates of a point on the line y = 4 , z = 0

coordinates of a point in the plane x + y – z = 0

direction numbers of a line in space

direction cosines of a line in space

Solution

Question 7

1/3

1/2

2/3

1/√3

Question 8

26

11 4/3

13

39

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90^o

30^o

60^o

0^o