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All questions of Three Dimensional Geometry for Class 10 Exam
The distance of the point (2, 3, – 5) from the plane x + 2y – 2z = 9 is:- a)2 units
- b)3/2 units
- c)3 units
- d)10/3 units
Correct answer is option 'C'. Can you explain this answer?
The distance of the point (2, 3, – 5) from the plane x + 2y – 2z = 9 is:
a)
2 units
b)
3/2 units
c)
3 units
d)
10/3 units
 | Nikita Singh answered |
Length of perpendicular from (2,3,-5) to the plane x + 2y − 2z − 9 = 0.
= |(2 + 2×3 −2×(−5) − 9)|√12 + 22 + (−2)2
= |2 + 6 + 10 − 9|/√9
= 9/3
= 3 units.
= |(2 + 2×3 −2×(−5) − 9)|√12 + 22 + (−2)2
= |2 + 6 + 10 − 9|/√9
= 9/3
= 3 units.
The equation of the plane passing through the point (3, – 3, 1) and perpendicular to the line joining the points (3, 4, – 1) and (2, – 1, 5) is:- a)– x – 5y + 6z + 18 = 0
- b)x – 5y + 6z + 18 = 0
- c)x + 5y – 6z + 18 = 0
- d)– x – 5y – 6z + 18 = 0
Correct answer is 'C'. Can you explain this answer?
The equation of the plane passing through the point (3, – 3, 1) and perpendicular to the line joining the points (3, 4, – 1) and (2, – 1, 5) is:
a)
– x – 5y + 6z + 18 = 0
b)
x – 5y + 6z + 18 = 0
c)
x + 5y – 6z + 18 = 0
d)
– x – 5y – 6z + 18 = 0
| | Leelu Bhai answered |
The ratio in which line joining (1, 2, 3) and (4, 5, 6) divide X-Y plane is ________- a)2
- b)-2
- c)1/2
- d)-1/2
Correct answer is option 'D'. Can you explain this answer?
The ratio in which line joining (1, 2, 3) and (4, 5, 6) divide X-Y plane is ________
a)
2
b)
-2
c)
1/2
d)
-1/2
 | Rajeev Kumar answered |
The coordinates of a point dividing the line segment joining (x1, y1, z1) and (x2, y2, z2) internally in the ratio m : n is 
Let ratio be k : 1.
So, z-coordinate of the point will be (k*6 + 1*3)/(k + 1).
We know, for X-Y plane, z coordinate is zero.
(6k + 1 * 3)/(k+1) = 0 ⇒ k = -1/2
Let ratio be k : 1.
So, z-coordinate of the point will be (k*6 + 1*3)/(k + 1).
We know, for X-Y plane, z coordinate is zero.
(6k + 1 * 3)/(k+1) = 0 ⇒ k = -1/2
Find midpoint of (1, 4, 6) and (5, 8, 10).- a)(6, 12, 8)
- b)(3, 6, 8)
- c)(1, 9, 12)
- d)(4, 9, 12)
Correct answer is option 'B'. Can you explain this answer?
Find midpoint of (1, 4, 6) and (5, 8, 10).
a)
(6, 12, 8)
b)
(3, 6, 8)
c)
(1, 9, 12)
d)
(4, 9, 12)
| | Rajeev Kumar answered |
We know, midpoint of (x1, y1, z1) and (x2, y2, z2) is (x1+x2) /2, (y1+y2) /2, (z1+z2)/2).
So, midpoint of (1, 4, 6) and (5, 8, 10) is ((1+5)/ 2, (4+8)/ 2, (6+10)/2) is (3, 6, 8).
So, midpoint of (1, 4, 6) and (5, 8, 10) is ((1+5)/ 2, (4+8)/ 2, (6+10)/2) is (3, 6, 8).
The three points A (1, 2, 3), B (3, 1, 2), C (2, 3, 1) form _________.- a)equilateral triangle
- b)right angled triangle
- c)isosceles triangle
- d)right angled isosceles triangle
Correct answer is option 'A'. Can you explain this answer?
The three points A (1, 2, 3), B (3, 1, 2), C (2, 3, 1) form _________.
a)
equilateral triangle
b)
right angled triangle
c)
isosceles triangle
d)
right angled isosceles triangle
 | Pathways Academy answered |
We know, distance between two points (x1, y1, z1) and (x2, y2, z2) is 
Since AB = BC = CA so, it forms equilateral triangle.
Since AB = BC = CA so, it forms equilateral triangle.
The length of the perpendicular from the origin to the plane 3x + 2y – 6z = 21 is:- a)3
- b)14
- c)21
- d)7
Correct answer is option 'A'. Can you explain this answer?
The length of the perpendicular from the origin to the plane 3x + 2y – 6z = 21 is:
a)
3
b)
14
c)
21
d)
7
| | Leelu Bhai answered |
Given equation of plane is : 3x + 2y - 6z - 21= 0
the length of perpendicular from a given point
(x' , y', z') on a plane ax + by + cz + d = 0 is given as :-
d = modulus of [{ax' + by' + cz' + d}/{√(a² + b² + c)²}]
so, d = modulus of [{(3*0) + (2*0) + (-6*0) + (-21)}/{√(3² + 2² + (-6)²)}]
d= modulus of (-21/√49) = (-21/7) = 3 units
hence option A is correct....
the length of perpendicular from a given point
(x' , y', z') on a plane ax + by + cz + d = 0 is given as :-
d = modulus of [{ax' + by' + cz' + d}/{√(a² + b² + c)²}]
so, d = modulus of [{(3*0) + (2*0) + (-6*0) + (-21)}/{√(3² + 2² + (-6)²)}]
d= modulus of (-21/√49) = (-21/7) = 3 units
hence option A is correct....
The Cartesian form of the equation of the plane 
is:- a)2x+3y-z=10
- b)2x+3y-z= √14
- c)2x+3y-z+√14=0
- d)2x+3y-z+10=0
Correct answer is option 'A'. Can you explain this answer?
The Cartesian form of the equation of the plane is:
is:a)
2x+3y-z=10
b)
2x+3y-z= √14
c)
2x+3y-z+√14=0
d)
2x+3y-z+10=0
 | Deepak Kapoor answered |
Let r = xi + yj + zk
(xi + yj + zk) . (2i + 3j - k) = 10
2x + 3y - k = 10
A and B be the points (3, 4, 5) and (-1, -3, -7), respectively, the equation of the set of points P such that PA2 + PB2 = k2, where k is a constant will- a)Have a term of k to the power 1 only
- b)Have a term of k to the power 2 only
- c)Have a term of k to the power 1 and another with k to the power 2
- d)Be independent of k
Correct answer is option 'B'. Can you explain this answer?
A and B be the points (3, 4, 5) and (-1, -3, -7), respectively, the equation of the set of points P such that PA2 + PB2 = k2, where k is a constant will
a)
Have a term of k to the power 1 only
b)
Have a term of k to the power 2 only
c)
Have a term of k to the power 1 and another with k to the power 2
d)
Be independent of k
| | Ravanlein Sekho answered |
D
The distance of the point (3, 4, 5) from X-axis is:- a)√41
- b)7
- c)2√11
- d)5√2
Correct answer is option 'A'. Can you explain this answer?
The distance of the point (3, 4, 5) from X-axis is:
a)
√41
b)
7
c)
2√11
d)
5√2
| | Sounak Yadav answered |
The distance of a point from the X-axis can be found by calculating the absolute value of its y-coordinate and z-coordinate.
In this case, the y-coordinate of the point is 4 and the z-coordinate is 5.
So, the distance of the point (3, 4, 5) from the X-axis is |4| + |5| = 4 + 5 = 9.
In this case, the y-coordinate of the point is 4 and the z-coordinate is 5.
So, the distance of the point (3, 4, 5) from the X-axis is |4| + |5| = 4 + 5 = 9.
Find the points on z-axis which are at a distance 
from the point (1, 2, 3).- a)(0, 0, 7), (0, 0, –1)
- b)(2, 7, 0), (–3, 2, 0)
- c)(1, 7, 0), (4, 3, 0)
- d)(0, 0, –7), (0, 0, 1)
Correct answer is option 'A'. Can you explain this answer?
Find the points on z-axis which are at a distance from the point (1, 2, 3).
from the point (1, 2, 3).a)
(0, 0, 7), (0, 0, –1)
b)
(2, 7, 0), (–3, 2, 0)
c)
(1, 7, 0), (4, 3, 0)
d)
(0, 0, –7), (0, 0, 1)
 | Yash Patel answered |
Let the point on Z axis be given as (0,0,z). The distance between (1,2,3) and (0,0,z) is given as [(1)2 + (2)2 + (3-z)2]½ = (21)1/2
5+(3−z)2=21
z2−6z−7=0
z=7,z = −1
Hence points are (0,0,7),(0,0,−1)
5+(3−z)2=21
z2−6z−7=0
z=7,z = −1
Hence points are (0,0,7),(0,0,−1)
The equation representing the set of points which are equidistant from the points (1, 2 , 3) and ( 3 , 2 , -1) is- a)2x – 2y = 0
- b)x – 2y = 0
- c)-x + 2y = 0
- d)x – 2z = 0
Correct answer is option 'D'. Can you explain this answer?
The equation representing the set of points which are equidistant from the points (1, 2 , 3) and ( 3 , 2 , -1) is
a)
2x – 2y = 0
b)
x – 2y = 0
c)
-x + 2y = 0
d)
x – 2z = 0
 | Arun Khanna answered |
Let P (x, y, z) be the point that is equidistant from points A(1, 2, 3) and B(3, 2, –1).
Accordingly, PA = PB
⇒ x2 – 2x + 1 + y2 – 4y + 4 + z2 – 6z + 9 = x2 – 6x + 9 + y2 – 4y + 4 + z2 + 2z + 1
⇒ –2x –4y – 6z + 14 = –6x – 4y + 2z + 14
⇒ – 2x – 6z + 6x – 2z = 0
⇒ 4x –8z = 0
⇒ x – 2z = 0
Thus, the required equation is x – 2z = 0.
In the following case, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them. 2x – 2y + 4z + 5 = 0 and 3x – 3y + 6z – 1 = 0- a)The planes are perpendicular
- b)The planes are parallel
- c)The planes are at 45∘
- d)The planes are at 55∘
Correct answer is option 'B'. Can you explain this answer?
In the following case, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them. 2x – 2y + 4z + 5 = 0 and 3x – 3y + 6z – 1 = 0
a)
The planes are perpendicular
b)
The planes are parallel
c)
The planes are at 45∘
d)
The planes are at 55∘
 | Pranavi Iyer answered |
The equation of the plane passing through the point (3, – 3, 1) and perpendicular to the line joining the points (3, 4, – 1) and (2, – 1, 5) is:- a)– x – 5y + 6z + 18 = 0
- b)x – 5y + 6z + 18 = 0
- c)x + 5y – 6z + 18 = 0
- d)– x – 5y – 6z + 18 = 0
Correct answer is option 'C'. Can you explain this answer?
The equation of the plane passing through the point (3, – 3, 1) and perpendicular to the line joining the points (3, 4, – 1) and (2, – 1, 5) is:
a)
– x – 5y + 6z + 18 = 0
b)
x – 5y + 6z + 18 = 0
c)
x + 5y – 6z + 18 = 0
d)
– x – 5y – 6z + 18 = 0
 | Mohit Rajpoot answered |
The equation of the plane passing through the point (3, – 3, 1) is:
a(x – 3) + b(y + 3) + c(z – 1) = 0 and the direction ratios of the line joining the points
(3, 4, – 1) and (2, – 1, 5) is 2 – 3, – 1 – 4, 5 + 1, i.e., – 1, – 5, 6.
Since the plane is perpendicular to the line whose direction ratios are – 1, – 5, 6, therefore, direction ratios of the normal to the plane is – 1, – 5, 6.
So, required equation of plane is: – 1(x – 3) – 5(y + 3) + 6(z – 1) = 0
i.e., x + 5y – 6z + 18 = 0.
a(x – 3) + b(y + 3) + c(z – 1) = 0 and the direction ratios of the line joining the points
(3, 4, – 1) and (2, – 1, 5) is 2 – 3, – 1 – 4, 5 + 1, i.e., – 1, – 5, 6.
Since the plane is perpendicular to the line whose direction ratios are – 1, – 5, 6, therefore, direction ratios of the normal to the plane is – 1, – 5, 6.
So, required equation of plane is: – 1(x – 3) – 5(y + 3) + 6(z – 1) = 0
i.e., x + 5y – 6z + 18 = 0.
The equation of the plane, which is at a distance of 5 unit from the origin and has 
as a normal vector, is:- a)
- b)
- c)
- d)
Correct answer is option 'D'. Can you explain this answer?
The equation of the plane, which is at a distance of 5 unit from the origin and has as a normal vector, is:
as a normal vector, is:a)
b)
c)
d)
 | Preeti Iyer answered |
x = 3i - 2j - 6k
|x| = ((3)2 + (2)2 + (6)2)
|x| = (49)½
|x| = 7
x = x/|x|
= (3i - 2j - 6k)/7
The required equation of plane is r.x = d
⇒ r.(3i - 2j - 6k)/7 = 5
⇒ r.(3i - 2j - 6k) = 35
|x| = ((3)2 + (2)2 + (6)2)
|x| = (49)½
|x| = 7
x = x/|x|
= (3i - 2j - 6k)/7
The required equation of plane is r.x = d
⇒ r.(3i - 2j - 6k)/7 = 5
⇒ r.(3i - 2j - 6k) = 35
If P (2, 3, 9), Q (2, 5, 5) and R (8, 5, 3) are vertices of a triangle then find the length of median through Q.- a)√24
- b)√38
- c)√11
- d)√53
Correct answer is option 'C'. Can you explain this answer?
If P (2, 3, 9), Q (2, 5, 5) and R (8, 5, 3) are vertices of a triangle then find the length of median through Q.
a)
√24
b)
√38
c)
√11
d)
√53
 | William Jenkins answered |
To find the length of the median through Q, we need to find the midpoint of PR and then calculate the distance between that midpoint and Q.
The midpoint of PR can be found by taking the average of the x-coordinates, the y-coordinates, and the z-coordinates of P and R.
Midpoint of PR = ((2+8)/2, (3+5)/2, (9+3)/2) = (5, 4, 6)
Now, we can calculate the distance between the midpoint of PR and Q using the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
Distance = sqrt((2-5)^2 + (5-4)^2 + (5-6)^2)
Distance = sqrt((-3)^2 + (1)^2 + (-1)^2)
Distance = sqrt(9 + 1 + 1)
Distance = sqrt(11)
Therefore, the length of the median through Q is sqrt(11).
The midpoint of PR can be found by taking the average of the x-coordinates, the y-coordinates, and the z-coordinates of P and R.
Midpoint of PR = ((2+8)/2, (3+5)/2, (9+3)/2) = (5, 4, 6)
Now, we can calculate the distance between the midpoint of PR and Q using the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
Distance = sqrt((2-5)^2 + (5-4)^2 + (5-6)^2)
Distance = sqrt((-3)^2 + (1)^2 + (-1)^2)
Distance = sqrt(9 + 1 + 1)
Distance = sqrt(11)
Therefore, the length of the median through Q is sqrt(11).
The coordinates of a point dividing the line segment joining (1, 2, 3) and (4, 5, 6) internally in the ratio 2:1 is _______.- a)(3, 4, 5)
- b)(5, 4, 3)
- c)(5, 3, 4)
- d)(4, 5, 3)
Correct answer is option 'A'. Can you explain this answer?
The coordinates of a point dividing the line segment joining (1, 2, 3) and (4, 5, 6) internally in the ratio 2:1 is _______.
a)
(3, 4, 5)
b)
(5, 4, 3)
c)
(5, 3, 4)
d)
(4, 5, 3)
| | Rajeev Kumar answered |
The coordinates of a point dividing the line segment joining (x1, y1, z1) and (x2, y2, z2) internally in the ratio m : n is 
So, the coordinates of a point dividing the line segment joining (1, 2, 3) and (4, 5, 6) internally in the ratio 2:1 is
So, the coordinates of a point dividing the line segment joining (1, 2, 3) and (4, 5, 6) internally in the ratio 2:1 is
In which ratio (3, 4, 5) divides the line segment joining (1, 2, 3) and (4, 5, 6) internally?- a)1:2
- b)2:1
- c)3:4
- d)4:3
Correct answer is option 'B'. Can you explain this answer?
In which ratio (3, 4, 5) divides the line segment joining (1, 2, 3) and (4, 5, 6) internally?
a)
1:2
b)
2:1
c)
3:4
d)
4:3
 | Isabella Martinez answered |
To determine the ratio in which the line segment joining two points divides internally, we can use the section formula. The section formula states that if a line segment joining two points (x1, y1, z1) and (x2, y2, z2) is divided by a point (x, y, z) internally in the ratio m:n, then the coordinates of the point (x, y, z) can be calculated using the following formula:
x = (m*x2 + n*x1) / (m + n)
y = (m*y2 + n*y1) / (m + n)
z = (m*z2 + n*z1) / (m + n)
Given that the two points are (1, 2, 3) and (4, 5, 6), and the ratio is 3:4:5, we can substitute these values into the section formula to find the coordinates of the point where the line segment is divided internally.
Calculating the coordinates:
x = (3*4 + 4*1) / (3 + 4) = (12 + 4) / 7 = 16/7
y = (3*5 + 4*2) / (3 + 4) = (15 + 8) / 7 = 23/7
z = (3*6 + 4*3) / (3 + 4) = (18 + 12) / 7 = 30/7
Therefore, the coordinates of the point where the line segment is divided internally are (16/7, 23/7, 30/7).
Now, we can calculate the distances between the two points and the point of division to confirm the ratio.
Distance between (1, 2, 3) and (16/7, 23/7, 30/7):
d1 = sqrt((16/7 - 1)^2 + (23/7 - 2)^2 + (30/7 - 3)^2)
= sqrt((9/7)^2 + (9/7)^2 + (9/7)^2)
= sqrt(81/49 + 81/49 + 81/49)
= sqrt(243/49)
= 9/7
Distance between (16/7, 23/7, 30/7) and (4, 5, 6):
d2 = sqrt((4 - 16/7)^2 + (5 - 23/7)^2 + (6 - 30/7)^2)
= sqrt((28/7 - 16/7)^2 + (35/7 - 23/7)^2 + (42/7 - 30/7)^2)
= sqrt((12/7)^2 + (12/7)^2 + (12/7)^2)
= sqrt(144/49 + 144/49 + 144/49)
= sqrt(432/49)
= 12/7
Therefore, the ratio in which the line segment is divided internally is d1:d2 = (9/7):(12/7) = 3:4, which matches option B.
x = (m*x2 + n*x1) / (m + n)
y = (m*y2 + n*y1) / (m + n)
z = (m*z2 + n*z1) / (m + n)
Given that the two points are (1, 2, 3) and (4, 5, 6), and the ratio is 3:4:5, we can substitute these values into the section formula to find the coordinates of the point where the line segment is divided internally.
Calculating the coordinates:
x = (3*4 + 4*1) / (3 + 4) = (12 + 4) / 7 = 16/7
y = (3*5 + 4*2) / (3 + 4) = (15 + 8) / 7 = 23/7
z = (3*6 + 4*3) / (3 + 4) = (18 + 12) / 7 = 30/7
Therefore, the coordinates of the point where the line segment is divided internally are (16/7, 23/7, 30/7).
Now, we can calculate the distances between the two points and the point of division to confirm the ratio.
Distance between (1, 2, 3) and (16/7, 23/7, 30/7):
d1 = sqrt((16/7 - 1)^2 + (23/7 - 2)^2 + (30/7 - 3)^2)
= sqrt((9/7)^2 + (9/7)^2 + (9/7)^2)
= sqrt(81/49 + 81/49 + 81/49)
= sqrt(243/49)
= 9/7
Distance between (16/7, 23/7, 30/7) and (4, 5, 6):
d2 = sqrt((4 - 16/7)^2 + (5 - 23/7)^2 + (6 - 30/7)^2)
= sqrt((28/7 - 16/7)^2 + (35/7 - 23/7)^2 + (42/7 - 30/7)^2)
= sqrt((12/7)^2 + (12/7)^2 + (12/7)^2)
= sqrt(144/49 + 144/49 + 144/49)
= sqrt(432/49)
= 12/7
Therefore, the ratio in which the line segment is divided internally is d1:d2 = (9/7):(12/7) = 3:4, which matches option B.
In the following case, determine whether the given planes are parallel orperpendicular, and in case they are neither, find the angles between them. 2x + y + 3z – 2 = 0 and x – 2y + 5 = 0- a)The planes are parallel
- b)The planes are at 45∘
- c)The planes are at 55∘
- d)The planes are perpendicular
Correct answer is option 'D'. Can you explain this answer?
In the following case, determine whether the given planes are parallel orperpendicular, and in case they are neither, find the angles between them. 2x + y + 3z – 2 = 0 and x – 2y + 5 = 0
a)
The planes are parallel
b)
The planes are at 45∘
c)
The planes are at 55∘
d)
The planes are perpendicular
 | Preethi Kaur answered |
We have ,
2x + y + 3z – 2 = 0 and x – 2y + 5 = 0. Let θ be the angle between the planes , then
2x + y + 3z – 2 = 0 and x – 2y + 5 = 0. Let θ be the angle between the planes , then
The points A (1, 2, -1), B (5, -2, 1), C (8, -7, 4), D (4, -3, 2) form __________.- a)trapezium
- b)rhombus
- c)square
- d)parallelogram
Correct answer is option 'D'. Can you explain this answer?
The points A (1, 2, -1), B (5, -2, 1), C (8, -7, 4), D (4, -3, 2) form __________.
a)
trapezium
b)
rhombus
c)
square
d)
parallelogram
| | Nova Brooks answered |
To determine the shape formed by the points A(1, 2, -1), B(5, -2, 1), C(8, -7, 4), and D(4, -3, 2), we need to analyze the properties of the quadrilateral formed by these points.
Step 1: Plotting the points
First, let's plot these points on a three-dimensional coordinate system to visualize their positions in space.
A(1, 2, -1) is located at coordinates (1, 2, -1).
B(5, -2, 1) is located at coordinates (5, -2, 1).
C(8, -7, 4) is located at coordinates (8, -7, 4).
D(4, -3, 2) is located at coordinates (4, -3, 2).
Step 2: Determining the sides
To determine the sides of the quadrilateral, we calculate the distances between each pair of consecutive points.
AB: Distance between A and B = √((5-1)^2 + (-2-2)^2 + (1-(-1))^2) = √(16 + 16 + 4) = √36 = 6
BC: Distance between B and C = √((8-5)^2 + (-7-(-2))^2 + (4-1)^2) = √(9 + 25 + 9) = √43
CD: Distance between C and D = √((4-8)^2 + (-3-(-7))^2 + (2-4)^2) = √(16 + 16 + 4) = √36 = 6
DA: Distance between D and A = √((1-4)^2 + (2-(-3))^2 + (-1-2)^2) = √(9 + 25 + 9) = √43
Step 3: Analyzing the sides
If a quadrilateral is a parallelogram, opposite sides must be equal in length.
In this case, AB = CD = 6 and BC = DA = √43, so the opposite sides are not equal. Therefore, the quadrilateral is not a parallelogram.
Step 4: Analyzing the angles
If a quadrilateral is a parallelogram, opposite angles must be equal in measure.
To determine the angles, we can use the dot product formula:
cosθ = (AB · BC) / (|AB| |BC|)
θ = angle between AB and BC
Using the dot product formula, we calculate:
AB · BC = (5-1)(8-5) + (-2-2)(-7-(-2)) + (1-(-1))(4-1)
= (4)(3) + (-4)(-5) + (2)(3)
= 12 + 20 + 6
= 38
|AB| = √36 = 6
|BC| = √43
cosθ = 38 / (6 √43)
Using a calculator, we find that cosθ ≈ 0.7709
Since the opposite angles are not equal,
Step 1: Plotting the points
First, let's plot these points on a three-dimensional coordinate system to visualize their positions in space.
A(1, 2, -1) is located at coordinates (1, 2, -1).
B(5, -2, 1) is located at coordinates (5, -2, 1).
C(8, -7, 4) is located at coordinates (8, -7, 4).
D(4, -3, 2) is located at coordinates (4, -3, 2).
Step 2: Determining the sides
To determine the sides of the quadrilateral, we calculate the distances between each pair of consecutive points.
AB: Distance between A and B = √((5-1)^2 + (-2-2)^2 + (1-(-1))^2) = √(16 + 16 + 4) = √36 = 6
BC: Distance between B and C = √((8-5)^2 + (-7-(-2))^2 + (4-1)^2) = √(9 + 25 + 9) = √43
CD: Distance between C and D = √((4-8)^2 + (-3-(-7))^2 + (2-4)^2) = √(16 + 16 + 4) = √36 = 6
DA: Distance between D and A = √((1-4)^2 + (2-(-3))^2 + (-1-2)^2) = √(9 + 25 + 9) = √43
Step 3: Analyzing the sides
If a quadrilateral is a parallelogram, opposite sides must be equal in length.
In this case, AB = CD = 6 and BC = DA = √43, so the opposite sides are not equal. Therefore, the quadrilateral is not a parallelogram.
Step 4: Analyzing the angles
If a quadrilateral is a parallelogram, opposite angles must be equal in measure.
To determine the angles, we can use the dot product formula:
cosθ = (AB · BC) / (|AB| |BC|)
θ = angle between AB and BC
Using the dot product formula, we calculate:
AB · BC = (5-1)(8-5) + (-2-2)(-7-(-2)) + (1-(-1))(4-1)
= (4)(3) + (-4)(-5) + (2)(3)
= 12 + 20 + 6
= 38
|AB| = √36 = 6
|BC| = √43
cosθ = 38 / (6 √43)
Using a calculator, we find that cosθ ≈ 0.7709
Since the opposite angles are not equal,
Find the equations of the planes that passes through three points (1, 1, 0), (1, 2, 1), (– 2, 2, – 1)- a)2x + 3y – 7z = 5
- b)2x + 5y – 3z = 5
- c)3x + 3y – 3z = 5
- d)2x + 3y – 3z = 5
Correct answer is option 'D'. Can you explain this answer?
Find the equations of the planes that passes through three points (1, 1, 0), (1, 2, 1), (– 2, 2, – 1)
a)
2x + 3y – 7z = 5
b)
2x + 5y – 3z = 5
c)
3x + 3y – 3z = 5
d)
2x + 3y – 3z = 5
 | Anand Khanna answered |
In cartesian co-ordinate system :
Equation of a plane passing through three non collinear
Equation of a plane passing through three non collinear
Points (x1, y1, z1) , (x2, y2, z2) and (x3, y3, z3) is given by :
Therefore, the equations of the planes that passes through three points (1,1,0), (1,2,1), (-2,2,-1) is given by :
⇒ (x-1)(-2) - (y-1) (3) + 3z = 0
⇒ 2x+3y - 3z = 5
Therefore, the equations of the planes that passes through three points (1,1,0), (1,2,1), (-2,2,-1) is given by :
⇒ (x-1)(-2) - (y-1) (3) + 3z = 0
⇒ 2x+3y - 3z = 5
If P (2, 3, 9), Q (2, 5, 5) and R (8, 5, 3) are vertices of a triangle then find the length of median through P.- a)√24
- b)√38
- c)√11
- d)√53
Correct answer is option 'B'. Can you explain this answer?
If P (2, 3, 9), Q (2, 5, 5) and R (8, 5, 3) are vertices of a triangle then find the length of median through P.
a)
√24
b)
√38
c)
√11
d)
√53
| | Rajeev Kumar answered |
We know, midpoint of (x1, y1, z1) and (x2, y2, z2) is ((x1+x2)/2, (y1+y2)/2, (z1+z2)/2).
Midpoint of line QR is (5, 5, 4).
Length of median through P is distance between midpoint of QR and P i.e.
Midpoint of line QR is (5, 5, 4).
Length of median through P is distance between midpoint of QR and P i.e.
The point equidistant from the points (0 , 0 , 0) , (1 , 0 , 0) , (0 , 2 , 0) , and (0 , 0 , 3) is- a)(- 1/2,- 1, - 3/2)
- b)(1 , 2 ,3)
- c)(1/2, 1, 3/2)
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
The point equidistant from the points (0 , 0 , 0) , (1 , 0 , 0) , (0 , 2 , 0) , and (0 , 0 , 3) is
a)
(- 1/2,- 1, - 3/2)
b)
(1 , 2 ,3)
c)
(1/2, 1, 3/2)
d)
none of these
| | Rahul Bansal answered |
The point equidistant from the given points must be the center of the sphere that contains these points. To find the center of this sphere, we can take the average of the coordinates of the four points.
The average of the x-coordinates is (0 + 1 + 0 + 0) / 4 = 1/4.
The average of the y-coordinates is (0 + 0 + 2 + 0) / 4 = 1/2.
The average of the z-coordinates is (0 + 0 + 0 + 3) / 4 = 3/4.
Therefore, the center of the sphere is at (1/4, 1/2, 3/4). This is the point equidistant from the four given points.
The correct answer is (c) (1/2, 1, 3/2).
Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0.- a)x + z + 2 = 0
- b)– x – z + 2 = 0
- c)x – z + 2 = 0
- d)x + z + 5 = 0
Correct answer is option 'C'. Can you explain this answer?
Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0.
a)
x + z + 2 = 0
b)
– x – z + 2 = 0
c)
x – z + 2 = 0
d)
x + z + 5 = 0
 | Avi Chawla answered |
The equation of the plane through the line of intersection of the planes
The three points A (3, 0, 3), B (5, 3, 2), C (6, 5, 5) form ________.- a)equilateral triangle
- b)right angled triangle
- c)isosceles triangle
- d)right angled isosceles triangle
Correct answer is option 'C'. Can you explain this answer?
The three points A (3, 0, 3), B (5, 3, 2), C (6, 5, 5) form ________.
a)
equilateral triangle
b)
right angled triangle
c)
isosceles triangle
d)
right angled isosceles triangle
| | Rajeev Kumar answered |
We know, distance between two points (x1, y1, z1) and (x2, y2, z2) is 
Since AB = BC so, it forms isosceles triangle.
Since AB = BC so, it forms isosceles triangle.
If coordinates of vertices of a triangle are (7, 6, 4), (5, 4, 6), (9, 5, 8), find the coordinates of centroid of the triangle.- a)(7, 5, 3)
- b)(7, 3, 5)
- c)(5, 3, 7)
- d)(3, 5, 7)
Correct answer is option 'A'. Can you explain this answer?
If coordinates of vertices of a triangle are (7, 6, 4), (5, 4, 6), (9, 5, 8), find the coordinates of centroid of the triangle.
a)
(7, 5, 3)
b)
(7, 3, 5)
c)
(5, 3, 7)
d)
(3, 5, 7)
| | Rajeev Kumar answered |
If coordinates of vertices of a triangle are (x1, y1, z1), (x2, y2, z2), (x3, y3, z3) the coordinates of centroid of the triangle are ((x1+x2+x3)/3, (y1+y2+y3)/3, (z1+z2+z3)/3)
So, coordinates of centroid of the given triangle are ((7 + 5 + 9)/3, (6 + 4 + 5)/3, (4 + 6 + 8)/3) = (7, 5, 3).
So, coordinates of centroid of the given triangle are ((7 + 5 + 9)/3, (6 + 4 + 5)/3, (4 + 6 + 8)/3) = (7, 5, 3).
The coordinates of a point dividing the line segment joining (1, 2, 3) and (4, 5, 6) externally in the ratio 2:1 is __________.- a)(4, 5, 6)
- b)(6, 8, 9)
- c)(7, 8, 9)
- d)(8, 6, 4)
Correct answer is option 'C'. Can you explain this answer?
The coordinates of a point dividing the line segment joining (1, 2, 3) and (4, 5, 6) externally in the ratio 2:1 is __________.
a)
(4, 5, 6)
b)
(6, 8, 9)
c)
(7, 8, 9)
d)
(8, 6, 4)
| | Rajeev Kumar answered |
The coordinates of a point dividing the line segment joining (x1, y1, z1) and (x2, y2, z2) externally in the ratio m : n is 
So, the coordinates of a point dividing the line segment joining (1, 2, 3) and (4, 5, 6) externally in the ratio 2:1 is
= (7, 8, 9).
So, the coordinates of a point dividing the line segment joining (1, 2, 3) and (4, 5, 6) externally in the ratio 2:1 is
= (7, 8, 9).Find the distance of the point (0, 0, 0) from the plane 3x – 4y + 12 z = 3- a)9/13
- b)7/13
- c)5/13
- d)3/13
Correct answer is option 'D'. Can you explain this answer?
Find the distance of the point (0, 0, 0) from the plane 3x – 4y + 12 z = 3
a)
9/13
b)
7/13
c)
5/13
d)
3/13
 | Krish Ghoshal answered |
As we know that the length of the perpendicular from point
P(x1,y1,z1) from the plane a1x+b1y+c1z+d1 = 0 is given by:
P(x1,y1,z1) from the plane a1x+b1y+c1z+d1 = 0 is given by:
The angle between two lines whose direction ratios are 1,2,1 and 2,-3,4 is:- a)30°
- b)60°
- c)90°
- d)45°
Correct answer is option 'C'. Can you explain this answer?
The angle between two lines whose direction ratios are 1,2,1 and 2,-3,4 is:
a)
30°
b)
60°
c)
90°
d)
45°
 | O Kaif Sid answered |
Cosx =( i+2j+k).(2i-3j+4k)/ √1+4+1 × √4+9+16
cosx= 2-6+4/√1+4+1 × √4+9+16
cosx = 0/√1+4+1 × √4+9+16
cosx = 0
therfore
x = 90
cosx= 2-6+4/√1+4+1 × √4+9+16
cosx = 0/√1+4+1 × √4+9+16
cosx = 0
therfore
x = 90
Angle between skew lines is- a)the angle between two non intersecting lines drawn from any point parallel to each of the skew lines
- b)the angle between two intersecting lines drawn from any point perpendicular to each of the skew lines
- c)the angle between two non intersecting lines drawn from any point anti – parallel to each of the skew lines
- d)the angle between two intersecting lines drawn from any point parallel to each of the skew lines
Correct answer is option 'D'. Can you explain this answer?
Angle between skew lines is
a)
the angle between two non intersecting lines drawn from any point parallel to each of the skew lines
b)
the angle between two intersecting lines drawn from any point perpendicular to each of the skew lines
c)
the angle between two non intersecting lines drawn from any point anti – parallel to each of the skew lines
d)
the angle between two intersecting lines drawn from any point parallel to each of the skew lines
 | Nilesh Goyal answered |
Angle between skew lines is the angle between two intersecting lines drawn from any point
What is the distance of the point (1, 2, 3) form the plane x – 3y + 2z + 13 = 0?- a)√14
- b)√13
- c)√11
- d)None of the above
Correct answer is option 'A'. Can you explain this answer?
What is the distance of the point (1, 2, 3) form the plane x – 3y + 2z + 13 = 0?
a)
√14
b)
√13
c)
√11
d)
None of the above
 | Bella Parker answered |
Distance of a point from a plane:
The distance between a point and a plane can be calculated using the formula:
\[ \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}} \]
Where (x1, y1, z1) is the coordinates of the point and Ax + By + Cz + D = 0 is the equation of the plane.
Given information:
Point P(1, 2, 3)
Equation of plane: x - 3y + 2z + 13 = 0
A = 1, B = -3, C = 2, D = 13
Calculating the distance:
Substitute the values into the formula:
\[ \frac{|1(1) + (-3)(2) + 2(3) + 13|}{\sqrt{1^2 + (-3)^2 + 2^2}} \]
\[ = \frac{|1 - 6 + 6 + 13|}{\sqrt{1 + 9 + 4}} \]
\[ = \frac{|14|}{\sqrt{14}} \]
\[ = \frac{14}{\sqrt{14}} \]
\[ = \sqrt{14} \]
Therefore, the distance of the point (1, 2, 3) from the plane x - 3y + 2z + 13 = 0 is \( \sqrt{14} \). Hence, option 'A' is the correct answer.
The distance between a point and a plane can be calculated using the formula:
\[ \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}} \]
Where (x1, y1, z1) is the coordinates of the point and Ax + By + Cz + D = 0 is the equation of the plane.
Given information:
Point P(1, 2, 3)
Equation of plane: x - 3y + 2z + 13 = 0
A = 1, B = -3, C = 2, D = 13
Calculating the distance:
Substitute the values into the formula:
\[ \frac{|1(1) + (-3)(2) + 2(3) + 13|}{\sqrt{1^2 + (-3)^2 + 2^2}} \]
\[ = \frac{|1 - 6 + 6 + 13|}{\sqrt{1 + 9 + 4}} \]
\[ = \frac{|14|}{\sqrt{14}} \]
\[ = \frac{14}{\sqrt{14}} \]
\[ = \sqrt{14} \]
Therefore, the distance of the point (1, 2, 3) from the plane x - 3y + 2z + 13 = 0 is \( \sqrt{14} \). Hence, option 'A' is the correct answer.
The three points A (7, 0, 10), B (6, -1, 6), C (9, -4, 6) form ________.- a)equilateral triangle
- b)right angled triangle
- c)isosceles triangle
- d)right angled isosceles triangle
Correct answer is option 'D'. Can you explain this answer?
The three points A (7, 0, 10), B (6, -1, 6), C (9, -4, 6) form ________.
a)
equilateral triangle
b)
right angled triangle
c)
isosceles triangle
d)
right angled isosceles triangle
| | Isabella Martinez answered |
Explanation:
Right Angled Isosceles Triangle:
- A right-angled isosceles triangle is a triangle with one right angle (90 degrees) and two equal sides.
- In this case, we can calculate the distances between the points to determine the type of triangle formed.
Calculating Distances:
- The distance formula between two points (x1, y1, z1) and (x2, y2, z2) in 3D space is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
Distance AB:
- AB = √((6 - 7)^2 + (-1 - 0)^2 + (6 - 10)^2)
- AB = √(1 + 1 + 16)
- AB = √18
Distance BC:
- BC = √((9 - 6)^2 + (-4 - (-1))^2 + (6 - 6)^2)
- BC = √(9 + 9 + 0)
- BC = √18
Distance AC:
- AC = √((9 - 7)^2 + (-4 - 0)^2 + (6 - 10)^2)
- AC = √(4 + 16 + 16)
- AC = √36
- AC = 6
Conclusion:
- Since AB = BC = √18 and AC = 6, the triangle formed by the points A, B, and C is a right-angled isosceles triangle.
- Therefore, the correct answer is option 'D'.
Right Angled Isosceles Triangle:
- A right-angled isosceles triangle is a triangle with one right angle (90 degrees) and two equal sides.
- In this case, we can calculate the distances between the points to determine the type of triangle formed.
Calculating Distances:
- The distance formula between two points (x1, y1, z1) and (x2, y2, z2) in 3D space is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
Distance AB:
- AB = √((6 - 7)^2 + (-1 - 0)^2 + (6 - 10)^2)
- AB = √(1 + 1 + 16)
- AB = √18
Distance BC:
- BC = √((9 - 6)^2 + (-4 - (-1))^2 + (6 - 6)^2)
- BC = √(9 + 9 + 0)
- BC = √18
Distance AC:
- AC = √((9 - 7)^2 + (-4 - 0)^2 + (6 - 10)^2)
- AC = √(4 + 16 + 16)
- AC = √36
- AC = 6
Conclusion:
- Since AB = BC = √18 and AC = 6, the triangle formed by the points A, B, and C is a right-angled isosceles triangle.
- Therefore, the correct answer is option 'D'.
Find the distance between two points (5, 6, 7) and (2, 6, 3).- a)3 units
- b)0 units
- c)4 units
- d)5 units
Correct answer is option 'D'. Can you explain this answer?
Find the distance between two points (5, 6, 7) and (2, 6, 3).
a)
3 units
b)
0 units
c)
4 units
d)
5 units
| | Rajeev Kumar answered |
We know, distance between two points (x1, y1, z1) and (x2, y2, z2) is 
So, distance between two points (5, 6, 7) and (2, 6, 3) will be
So, distance between two points (5, 6, 7) and (2, 6, 3) will be
Find the points which trisects the line joining (4, 9, 8) and (13, 27, -4).- a)(0, 21, 10)
- b)(0, 21, 4)
- c)(10, 21, 0)
- d)(4, 4, 0)
Correct answer is option 'C'. Can you explain this answer?
Find the points which trisects the line joining (4, 9, 8) and (13, 27, -4).
a)
(0, 21, 10)
b)
(0, 21, 4)
c)
(10, 21, 0)
d)
(4, 4, 0)
| | Rajeev Kumar answered |
Points which trisect the line divides it into 2:1 and 1:2.
The coordinates of a point dividing the line segment joining (x1, y1, z1) and (x2, y2, z2) internally in the ratio m : n is
For 1:2, coordinates of point are
= (7, 15, 4)
For 2:1, coordinates of point are
= (10, 21, 0)
The coordinates of a point dividing the line segment joining (x1, y1, z1) and (x2, y2, z2) internally in the ratio m : n is
For 1:2, coordinates of point are
= (7, 15, 4)For 2:1, coordinates of point are
= (10, 21, 0)The points A (0 , 0 , 0) , B (1 , √3 , 0) , C (2 , 0 , 0) and D (1 , 0 , √3) are the vertices of- a)parallelogram
- b)square
- c)rhombus
- d)none of these
Correct answer is option 'D'. Can you explain this answer?
The points A (0 , 0 , 0) , B (1 , √3 , 0) , C (2 , 0 , 0) and D (1 , 0 , √3) are the vertices of
a)
parallelogram
b)
square
c)
rhombus
d)
none of these
 | Jay Khanna answered |
Given Points: A (0, 0, 0), B (1, 3, 0), C (2, 0, 0), D (1, 0, 3)
Checking for a parallelogram:
- A parallelogram is a quadrilateral with opposite sides parallel to each other.
- AB is not parallel to CD since AB has a slope of 3 and CD has a slope of -3/2.
- AD is not parallel to BC since AD has a slope of -3 and BC has a slope of 3/2.
- Therefore, the given points do not form a parallelogram.
Checking for a square:
- A square is a quadrilateral with all sides equal in length and all angles equal to 90 degrees.
- AB and CD have a length of sqrt(10) but AD and BC have a length of 3.
- Therefore, the given points do not form a square.
Checking for a rhombus:
- A rhombus is a quadrilateral with all sides equal in length but opposite angles are not necessarily equal to 90 degrees.
- AB and CD have a length of sqrt(10) but AD and BC have a length of 3.
- Therefore, the given points do not form a rhombus.
Conclusion:
Since the given points do not form a parallelogram, square, or rhombus, the correct answer is option 'D' (none of these).
Checking for a parallelogram:
- A parallelogram is a quadrilateral with opposite sides parallel to each other.
- AB is not parallel to CD since AB has a slope of 3 and CD has a slope of -3/2.
- AD is not parallel to BC since AD has a slope of -3 and BC has a slope of 3/2.
- Therefore, the given points do not form a parallelogram.
Checking for a square:
- A square is a quadrilateral with all sides equal in length and all angles equal to 90 degrees.
- AB and CD have a length of sqrt(10) but AD and BC have a length of 3.
- Therefore, the given points do not form a square.
Checking for a rhombus:
- A rhombus is a quadrilateral with all sides equal in length but opposite angles are not necessarily equal to 90 degrees.
- AB and CD have a length of sqrt(10) but AD and BC have a length of 3.
- Therefore, the given points do not form a rhombus.
Conclusion:
Since the given points do not form a parallelogram, square, or rhombus, the correct answer is option 'D' (none of these).
A, B, C and D are four points in spaces such that AB = BC = CD = DA. Then ABCD is a- a)skew quadrilateral
- b)rhombus
- c)nothing can be said
- d)rectangle
Correct answer is option 'C'. Can you explain this answer?
A, B, C and D are four points in spaces such that AB = BC = CD = DA. Then ABCD is a
a)
skew quadrilateral
b)
rhombus
c)
nothing can be said
d)
rectangle
 | Meghana Pillai answered |
Explanation:
Given:
- Points A, B, C, and D are such that AB = BC = CD = DA.
Analysis:
- When all four sides of a quadrilateral are equal, it doesn't necessarily mean that the quadrilateral is a special type like a rectangle, rhombus, or square.
- In this case, since only the side lengths are given, we cannot determine the angles between the sides. Therefore, we cannot conclude whether ABCD is a rhombus, rectangle, or any other specific type of quadrilateral.
Conclusion:
- Without additional information about the angles or other properties of the quadrilateral, we cannot definitively say that ABCD is a specific type of quadrilateral.
- Therefore, the correct answer is option 'C' - nothing can be said.
A point is 5 units away from the vertical plane and 4 units away from profile plane and 3 units away from horizontal plane in 1st quadrant then the projections are drawn on paper the distance between the front view and top view of point is _____________- a)7 units
- b)8 units
- c)9 units
- d)5 units
Correct answer is option 'B'. Can you explain this answer?
A point is 5 units away from the vertical plane and 4 units away from profile plane and 3 units away from horizontal plane in 1st quadrant then the projections are drawn on paper the distance between the front view and top view of point is _____________
a)
7 units
b)
8 units
c)
9 units
d)
5 units
 | Kavita Joshi answered |
Since the point is 3 units away from the horizontal plane the distance from the point to xy reference line will be 3 units. And then the point is at distance of 5 units from the vertical plane the distance from reference line and point will be 5, sum is 8.
A(4,7,8) B(2,3,4) , C (-1,-2,1) and D(1,2,5) are vertices of a quadrilateral. The quadrilateral is a- a)Rhombus
- b)Rectangle
- c)Square
- d)Parallelogram
Correct answer is option 'D'. Can you explain this answer?
A(4,7,8) B(2,3,4) , C (-1,-2,1) and D(1,2,5) are vertices of a quadrilateral. The quadrilateral is a
a)
Rhombus
b)
Rectangle
c)
Square
d)
Parallelogram
| | Surbhi Bose answered |
AB = [(2−4)2 +(3−2)2 +(4−8)2]1/2
AB= [(−2)2 + (1)2 + (−4)2]^1/2
AB = (21)1/2
Similarly you find that BC= (43)1/2
CD= (33)1/2 and DA= (43)1/2
Hence opposite sides of quadrilateral are equal, Now we check the diagonals
AC= [(-1-4)2 + (−2-7)2 + (1−8)2]1/2
AC= (155)1/2
similarly BD= (3)1/2
Diagonals are not equal
direction ratio of line passing through AB is (-2,-4,-4)
direction ratio of line passing through CD is (2,4,4), As the dr of AB and CD are proportional which means AB is parallel to CD,
Similarly check for BC and DA then you will find that they are also parallel
Hence it is parallelogram.
AB= [(−2)2 + (1)2 + (−4)2]^1/2
AB = (21)1/2
Similarly you find that BC= (43)1/2
CD= (33)1/2 and DA= (43)1/2
Hence opposite sides of quadrilateral are equal, Now we check the diagonals
AC= [(-1-4)2 + (−2-7)2 + (1−8)2]1/2
AC= (155)1/2
similarly BD= (3)1/2
Diagonals are not equal
direction ratio of line passing through AB is (-2,-4,-4)
direction ratio of line passing through CD is (2,4,4), As the dr of AB and CD are proportional which means AB is parallel to CD,
Similarly check for BC and DA then you will find that they are also parallel
Hence it is parallelogram.
The points (1, -1, 3), (2, -4, 5) and (5, -13, 11) are:- a)Vertices of a right triangle
- b)Vertices of a square
- c)Collinear
- d)Coplanar
Correct answer is option 'C'. Can you explain this answer?
The points (1, -1, 3), (2, -4, 5) and (5, -13, 11) are:
a)
Vertices of a right triangle
b)
Vertices of a square
c)
Collinear
d)
Coplanar
| | Soumya Nambiar answered |
Given Points:
The given points are:
- (1, -1, 3)
- (2, -4, 5)
- (5, -13, 11)
Checking for Collinearity:
To check if the points are collinear, we need to see if they lie on the same line. We can use the concept of slopes to determine this.
Finding Slopes:
Let's find the slopes between the first two points and the first and third points.
- Slope between (1, -1, 3) and (2, -4, 5):
m1 = (y2 - y1) / (x2 - x1) = (-4 - (-1)) / (2 - 1) = -3 / 1 = -3
- Slope between (1, -1, 3) and (5, -13, 11):
m2 = (y2 - y1) / (x2 - x1) = (-13 - (-1)) / (5 - 1) = -12 / 4 = -3
Comparing Slopes:
Since both slopes m1 and m2 are equal to -3, it implies that all three points lie on the same line. Therefore, the given points are collinear.
Explanation:
Collinear points are the points that lie on the same straight line. In this case, the three given points lie on the same line, so they are collinear. This can be visually represented by imagining a line passing through the three points.
Therefore, the correct answer is option 'C' - Collinear.
The given points are:
- (1, -1, 3)
- (2, -4, 5)
- (5, -13, 11)
Checking for Collinearity:
To check if the points are collinear, we need to see if they lie on the same line. We can use the concept of slopes to determine this.
Finding Slopes:
Let's find the slopes between the first two points and the first and third points.
- Slope between (1, -1, 3) and (2, -4, 5):
m1 = (y2 - y1) / (x2 - x1) = (-4 - (-1)) / (2 - 1) = -3 / 1 = -3
- Slope between (1, -1, 3) and (5, -13, 11):
m2 = (y2 - y1) / (x2 - x1) = (-13 - (-1)) / (5 - 1) = -12 / 4 = -3
Comparing Slopes:
Since both slopes m1 and m2 are equal to -3, it implies that all three points lie on the same line. Therefore, the given points are collinear.
Explanation:
Collinear points are the points that lie on the same straight line. In this case, the three given points lie on the same line, so they are collinear. This can be visually represented by imagining a line passing through the three points.
Therefore, the correct answer is option 'C' - Collinear.
Equation of a plane which is at a distance d from the origin and the direction cosines of the normal to the plane are l, m, n is.- a)lx – my + nz = d
- b)– lx + my + nz = d
- c)lx + my + nz = d
- d)lx + my + nz = – d
Correct answer is option 'C'. Can you explain this answer?
Equation of a plane which is at a distance d from the origin and the direction cosines of the normal to the plane are l, m, n is.
a)
lx – my + nz = d
b)
– lx + my + nz = d
c)
lx + my + nz = d
d)
lx + my + nz = – d
 | Aravind Nambiar answered |
In Cartesian co – ordinate system Equation of a plane which is at a distance d from the origin and the direction cosines of the normal to the plane are l, m, n is given by : lx + my + nz = d.
Find the points which trisects the line joining (4, 9, 8) and (13, 27, -4).- a)(7, 4, 15)
- b)(7, 15, 4)
- c)(4, 15, 7)
- d)(4, 7, 15)
Correct answer is option 'B'. Can you explain this answer?
Find the points which trisects the line joining (4, 9, 8) and (13, 27, -4).
a)
(7, 4, 15)
b)
(7, 15, 4)
c)
(4, 15, 7)
d)
(4, 7, 15)
| | Stella Hayes answered |
Identifying the Trisection Points:
To find the points which trisect the line joining (4, 9, 8) and (13, 27, -4), we need to divide the line into three equal parts.
Calculating the trisection points:
1. First, calculate the differences between the coordinates of the two given points:
- Difference in x-coordinates: 13 - 4 = 9
- Difference in y-coordinates: 27 - 9 = 18
- Difference in z-coordinates: -4 - 8 = -12
2. Divide these differences by 3 to get the increments for each trisection point:
- Increment in x: 9 / 3 = 3
- Increment in y: 18 / 3 = 6
- Increment in z: -12 / 3 = -4
3. Starting from the first point (4, 9, 8), add the increments to find the trisection points:
- First trisection point: (4 + 3, 9 + 6, 8 - 4) = (7, 15, 4)
- Second trisection point: (4 + 2*3, 9 + 2*6, 8 - 2*4) = (10, 21, 0)
- Third trisection point: (4 + 3*3, 9 + 3*6, 8 - 3*4) = (13, 27, -4)
Therefore, the trisection points are (7, 15, 4), (10, 21, 0), and (13, 27, -4). The correct answer is option (B) (7, 15, 4).
To find the points which trisect the line joining (4, 9, 8) and (13, 27, -4), we need to divide the line into three equal parts.
Calculating the trisection points:
1. First, calculate the differences between the coordinates of the two given points:
- Difference in x-coordinates: 13 - 4 = 9
- Difference in y-coordinates: 27 - 9 = 18
- Difference in z-coordinates: -4 - 8 = -12
2. Divide these differences by 3 to get the increments for each trisection point:
- Increment in x: 9 / 3 = 3
- Increment in y: 18 / 3 = 6
- Increment in z: -12 / 3 = -4
3. Starting from the first point (4, 9, 8), add the increments to find the trisection points:
- First trisection point: (4 + 3, 9 + 6, 8 - 4) = (7, 15, 4)
- Second trisection point: (4 + 2*3, 9 + 2*6, 8 - 2*4) = (10, 21, 0)
- Third trisection point: (4 + 3*3, 9 + 3*6, 8 - 3*4) = (13, 27, -4)
Therefore, the trisection points are (7, 15, 4), (10, 21, 0), and (13, 27, -4). The correct answer is option (B) (7, 15, 4).
The lines l1 and l2 intersect. The shortest distance between them is- a)infinity
- b)negative
- c)zero
- d)positive
Correct answer is option 'C'. Can you explain this answer?
The lines l1 and l2 intersect. The shortest distance between them is
a)
infinity
b)
negative
c)
zero
d)
positive
| | Khaja Moinuddin answered |
If two lines are intersecting then the two lines will definitely have a point on common I.e there will be a point on L1 which is also a point on L2 then the least distance possible would be the distance between the points which are common to both L1 and L2 I.e. zero....so the answer for this question is zero
Find the cartesian equation of the plane which passes through the point (5, 2, -4) and perpendicular to the line with direction ratios 2, 3, -1?- a)x + 2y - 3z = 18
- b)2x + 3y - z = 20
- c)2x + 3y - 5y = 24
- d)2x + 5y - 7z = 15
Correct answer is option 'B'. Can you explain this answer?
Find the cartesian equation of the plane which passes through the point (5, 2, -4) and perpendicular to the line with direction ratios 2, 3, -1?
a)
x + 2y - 3z = 18
b)
2x + 3y - z = 20
c)
2x + 3y - 5y = 24
d)
2x + 5y - 7z = 15
| | Rajeev Kumar answered |
Given:
The plane which passes through the point (5, 2, -4)
Plane is perpendicular to the line with direction ratios 2, 3, -1
The plane which passes through the point (5, 2, -4)
Plane is perpendicular to the line with direction ratios 2, 3, -1
Concept:
Cartesian equation of the plane passing through (x1 , y1 , z1) and perpendicular to line having drs a, b, c is given by :
a(x - x1) + b(y - y1) + c(z - z1) = 0
Cartesian equation of the plane passing through (x1 , y1 , z1) and perpendicular to line having drs a, b, c is given by :
a(x - x1) + b(y - y1) + c(z - z1) = 0
Solution:
Equation of plane :
2(x - 5) + 3(y - 2) -(z - (-4)) = 0
⇒ 2x + 3y - z = 20
Equation of plane :
2(x - 5) + 3(y - 2) -(z - (-4)) = 0
⇒ 2x + 3y - z = 20
Find the values of p so that the lines 
are at right angles.- a)p =72/15
- b)p =70/11
- c)p =70/12
- d)p =71/13
Correct answer is option 'B'. Can you explain this answer?
Find the values of p so that the lines are at right angles.
are at right angles.a)
p =72/15
b)
p =70/11
c)
p =70/12
d)
p =71/13
| | Avantika Saha answered |
Give lines are :
and 
The D.R's of the lines are -3, 2p/7, 2 and -3p/7, 1, -5
and The D.R's of the lines are -3, 2p/7, 2 and -3p/7, 1, -5
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