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If a point R(4, y, z) lies on the line segment joining the points P (2,-3, 4) and Q(8, 0, 10), then the distance of R from the origin is:
- a)2√14
- b)6
- c)2√21
- d)√53
Correct answer is option 'A'. Can you explain this answer?
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If a point R(4, y, z) lies on the line segment joining the points P (2...
Here, P, Q, R are collinear ⇒ 
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If a point R(4, y, z) lies on the line segment joining the points P (2...
The line segment joining P(2,-3,4) and Q(8,0,10) can be parameterized as follows:
x = 2 + t(8-2)
y = -3 + t(0-(-3))
z = 4 + t(10-4)
Simplifying these equations, we get:
x = 2 + 6t
y = -3 + 3t
z = 4 + 6t
Now we substitute the coordinates of point R(4,y,z) into these equations:
4 = 2 + 6t
y = -3 + 3t
z = 4 + 6t
Solving the first equation, we find t = 1/3.
Substituting this value of t into the second and third equations, we get:
y = -3 + 3(1/3) = 0
z = 4 + 6(1/3) = 6
So the coordinates of R are (4,0,6).
The distance of R from the origin can be found using the distance formula:
distance = √((x-0)^2 + (y-0)^2 + (z-0)^2)
= √(4^2 + 0^2 + 6^2)
= √(16 + 0 + 36)
= √52
≈ 7.211
Therefore, the distance of R from the origin is approximately 7.211.
x = 2 + t(8-2)
y = -3 + t(0-(-3))
z = 4 + t(10-4)
Simplifying these equations, we get:
x = 2 + 6t
y = -3 + 3t
z = 4 + 6t
Now we substitute the coordinates of point R(4,y,z) into these equations:
4 = 2 + 6t
y = -3 + 3t
z = 4 + 6t
Solving the first equation, we find t = 1/3.
Substituting this value of t into the second and third equations, we get:
y = -3 + 3(1/3) = 0
z = 4 + 6(1/3) = 6
So the coordinates of R are (4,0,6).
The distance of R from the origin can be found using the distance formula:
distance = √((x-0)^2 + (y-0)^2 + (z-0)^2)
= √(4^2 + 0^2 + 6^2)
= √(16 + 0 + 36)
= √52
≈ 7.211
Therefore, the distance of R from the origin is approximately 7.211.
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If a point R(4, y, z) lies on the line segment joining the points P (2,-3, 4) and Q(8, 0, 10), then the distance of R from the origin is:a)2√14b)6c)2√21d)√53Correct answer is option 'A'. Can you explain this answer?
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