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If the distance from point P(1,1,1) to the line passing through the points Q(0,6,8)andR(-1,4,7) is expressed in the form root over p/q (1/2^p/q) where p and q are coprime ,then the value of (p+q)(p+q-1)/2equals:?
Most Upvoted Answer
If the distance from point P(1,1,1) to the line passing through the po...
Approach:
To find the distance between the point P and the line QR, we will use the formula for the distance between a point and a line in 3D space.
We will first find the direction vector of the line QR by subtracting the coordinates of Q from R.
Then we will find a vector from point P to any point on the line QR.
We will take the cross product of these two vectors to get the perpendicular distance between point P and the line QR.
Finally, we will divide the magnitude of this vector by the magnitude of the direction vector of the line QR to get the distance.
We will simplify the expression and express it in the given form.
Solution:
Finding the direction vector of the line QR:
Let's subtract the coordinates of Q from R to get the direction vector of the line QR.
QR = R - Q
= (-1 - 0, 4 - 6, 7 - 8)
= (-1, -2, -1)
Finding a vector from point P to any point on the line QR:
Let's take the vector from point Q to point P.
PQ = P - Q
= (1 - 0, 1 - 6, 1 - 8)
= (1, -5, -7)
Finding the perpendicular distance between point P and the line QR:
Let's take the cross product of the direction vector of the line QR and the vector from point P to any point on the line QR.
N = QR x PQ
= (-1, -2, -1) x (1, -5, -7)
= (9, 6, -3)
The magnitude of the vector N is:
|N| = sqrt(9^2 + 6^2 + (-3)^2)
= sqrt(126)
Finding the distance:
The magnitude of the direction vector of the line QR is:
|QR| = sqrt((-1)^2 + (-2)^2 + (-1)^2)
= sqrt(6)
The distance between point P and the line QR is:
d = |N| / |QR|
= sqrt(126) / sqrt(6)
= sqrt(21/2)
Now, let's simplify the expression in the given form.
sqrt(21/2) = sqrt(21) / sqrt(2)
= sqrt(21) / 2^(1/2)
= sqrt(21) / 2^(1/2) * 2^(1/2) / 2^(1/2)
= sqrt(21) * 2^(1/2) / 2
= (sqrt(21) * 2^(1/2)) / (2^1)
Therefore, p = 21 and q = 2^2 = 4.
(pq)(p q-1)/2 = (21*3)/2 = 31.5
Answer:
The value of (p q)(p q-1)/2 is 31.5.
To find the distance between the point P and the line QR, we will use the formula for the distance between a point and a line in 3D space.
We will first find the direction vector of the line QR by subtracting the coordinates of Q from R.
Then we will find a vector from point P to any point on the line QR.
We will take the cross product of these two vectors to get the perpendicular distance between point P and the line QR.
Finally, we will divide the magnitude of this vector by the magnitude of the direction vector of the line QR to get the distance.
We will simplify the expression and express it in the given form.
Solution:
Finding the direction vector of the line QR:
Let's subtract the coordinates of Q from R to get the direction vector of the line QR.
QR = R - Q
= (-1 - 0, 4 - 6, 7 - 8)
= (-1, -2, -1)
Finding a vector from point P to any point on the line QR:
Let's take the vector from point Q to point P.
PQ = P - Q
= (1 - 0, 1 - 6, 1 - 8)
= (1, -5, -7)
Finding the perpendicular distance between point P and the line QR:
Let's take the cross product of the direction vector of the line QR and the vector from point P to any point on the line QR.
N = QR x PQ
= (-1, -2, -1) x (1, -5, -7)
= (9, 6, -3)
The magnitude of the vector N is:
|N| = sqrt(9^2 + 6^2 + (-3)^2)
= sqrt(126)
Finding the distance:
The magnitude of the direction vector of the line QR is:
|QR| = sqrt((-1)^2 + (-2)^2 + (-1)^2)
= sqrt(6)
The distance between point P and the line QR is:
d = |N| / |QR|
= sqrt(126) / sqrt(6)
= sqrt(21/2)
Now, let's simplify the expression in the given form.
sqrt(21/2) = sqrt(21) / sqrt(2)
= sqrt(21) / 2^(1/2)
= sqrt(21) / 2^(1/2) * 2^(1/2) / 2^(1/2)
= sqrt(21) * 2^(1/2) / 2
= (sqrt(21) * 2^(1/2)) / (2^1)
Therefore, p = 21 and q = 2^2 = 4.
(pq)(p q-1)/2 = (21*3)/2 = 31.5
Answer:
The value of (p q)(p q-1)/2 is 31.5.
Community Answer
If the distance from point P(1,1,1) to the line passing through the po...
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If the distance from point P(1,1,1) to the line passing through the points Q(0,6,8)andR(-1,4,7) is expressed in the form root over p/q (1/2^p/q) where p and q are coprime ,then the value of (p+q)(p+q-1)/2equals:?
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If the distance from point P(1,1,1) to the line passing through the points Q(0,6,8)andR(-1,4,7) is expressed in the form root over p/q (1/2^p/q) where p and q are coprime ,then the value of (p+q)(p+q-1)/2equals:? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If the distance from point P(1,1,1) to the line passing through the points Q(0,6,8)andR(-1,4,7) is expressed in the form root over p/q (1/2^p/q) where p and q are coprime ,then the value of (p+q)(p+q-1)/2equals:? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the distance from point P(1,1,1) to the line passing through the points Q(0,6,8)andR(-1,4,7) is expressed in the form root over p/q (1/2^p/q) where p and q are coprime ,then the value of (p+q)(p+q-1)/2equals:?.
If the distance from point P(1,1,1) to the line passing through the points Q(0,6,8)andR(-1,4,7) is expressed in the form root over p/q (1/2^p/q) where p and q are coprime ,then the value of (p+q)(p+q-1)/2equals:? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If the distance from point P(1,1,1) to the line passing through the points Q(0,6,8)andR(-1,4,7) is expressed in the form root over p/q (1/2^p/q) where p and q are coprime ,then the value of (p+q)(p+q-1)/2equals:? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the distance from point P(1,1,1) to the line passing through the points Q(0,6,8)andR(-1,4,7) is expressed in the form root over p/q (1/2^p/q) where p and q are coprime ,then the value of (p+q)(p+q-1)/2equals:?.
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