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A line with positive direction cosines passes through the point P(2, –1, 2) and makes equal angles with the coordinate axes. The line meets the plane 2x + y + z = 9 at point Q. The length of the line segment PQ equals
- a)1
- b)√2
- c)√3
- d)2
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
A line with positive direction cosines passes through the point P(2, &...
The line has +ve and equal direction cosines, these are 
or direction ratios are 1, 1, 1. Also thelines passes through P (2, – 1, 2).
∴ Equation of line is
or direction ratios are 1, 1, 1. Also thelines passes through P (2, – 1, 2).∴ Equation of line is
be a point on this line where it meets the plane 2 x + y + z = 9Then Q must satisfy the eqn of plane
∴ Q has coordintes (3, 0, 3)
Most Upvoted Answer
A line with positive direction cosines passes through the point P(2, &...
Given Information:
Point P(2, -1, 2)
Line with positive direction cosines passing through P
Line makes equal angles with coordinate axes
Plane 2x + y + z = 9
Point of intersection with plane is Q
Solution:
Step 1: Finding Direction Ratios of the Line
Since the line makes equal angles with the coordinate axes, the direction cosines will be equal.
Let the direction cosines be (l, l, l), where l is a constant.
Since the line passes through point P(2, -1, 2), we can write:
2/l = -1/l = 2/l
This gives us l = ±√3
Step 2: Finding Point Q
The line intersects the plane at point Q, so the coordinates of Q satisfy the equation of the plane:
2x + y + z = 9
Substitute the coordinates of Q as (2√3t, -√3t, 2√3t) where t is a parameter.
Plug these values into the equation of the plane and solve for t:
2(2√3t) + (-√3t) + 2√3t = 9
4√3t - √3t + 2√3t = 9
5√3t = 9
t = 9/(5√3) = √3/5
Step 3: Calculating Length of PQ
The coordinates of Q are (2√3(√3/5), -√3(√3/5), 2√3(√3/5)) = (6/5, -3/5, 6/5)
Using the distance formula, the length of PQ = √[(6-2)^2 + (-3+1)^2 + (6-2)^2] = √(16 + 4 + 16) = √36 = √3
Therefore, the length of the line segment PQ is √3, which corresponds to option 'c'.
Point P(2, -1, 2)
Line with positive direction cosines passing through P
Line makes equal angles with coordinate axes
Plane 2x + y + z = 9
Point of intersection with plane is Q
Solution:
Step 1: Finding Direction Ratios of the Line
Since the line makes equal angles with the coordinate axes, the direction cosines will be equal.
Let the direction cosines be (l, l, l), where l is a constant.
Since the line passes through point P(2, -1, 2), we can write:
2/l = -1/l = 2/l
This gives us l = ±√3
Step 2: Finding Point Q
The line intersects the plane at point Q, so the coordinates of Q satisfy the equation of the plane:
2x + y + z = 9
Substitute the coordinates of Q as (2√3t, -√3t, 2√3t) where t is a parameter.
Plug these values into the equation of the plane and solve for t:
2(2√3t) + (-√3t) + 2√3t = 9
4√3t - √3t + 2√3t = 9
5√3t = 9
t = 9/(5√3) = √3/5
Step 3: Calculating Length of PQ
The coordinates of Q are (2√3(√3/5), -√3(√3/5), 2√3(√3/5)) = (6/5, -3/5, 6/5)
Using the distance formula, the length of PQ = √[(6-2)^2 + (-3+1)^2 + (6-2)^2] = √(16 + 4 + 16) = √36 = √3
Therefore, the length of the line segment PQ is √3, which corresponds to option 'c'.
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A line with positive direction cosines passes through the point P(2, –1, 2) and makes equal angles with the coordinate axes. The line meets the plane2x + y + z = 9at point Q. The length of the line segment PQ equalsa)1b)√2c)√3d)2Correct answer is option 'C'. Can you explain this answer?
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A line with positive direction cosines passes through the point P(2, –1, 2) and makes equal angles with the coordinate axes. The line meets the plane2x + y + z = 9at point Q. The length of the line segment PQ equalsa)1b)√2c)√3d)2Correct answer is option 'C'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A line with positive direction cosines passes through the point P(2, –1, 2) and makes equal angles with the coordinate axes. The line meets the plane2x + y + z = 9at point Q. The length of the line segment PQ equalsa)1b)√2c)√3d)2Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A line with positive direction cosines passes through the point P(2, –1, 2) and makes equal angles with the coordinate axes. The line meets the plane2x + y + z = 9at point Q. The length of the line segment PQ equalsa)1b)√2c)√3d)2Correct answer is option 'C'. Can you explain this answer?.
A line with positive direction cosines passes through the point P(2, –1, 2) and makes equal angles with the coordinate axes. The line meets the plane2x + y + z = 9at point Q. The length of the line segment PQ equalsa)1b)√2c)√3d)2Correct answer is option 'C'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A line with positive direction cosines passes through the point P(2, –1, 2) and makes equal angles with the coordinate axes. The line meets the plane2x + y + z = 9at point Q. The length of the line segment PQ equalsa)1b)√2c)√3d)2Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A line with positive direction cosines passes through the point P(2, –1, 2) and makes equal angles with the coordinate axes. The line meets the plane2x + y + z = 9at point Q. The length of the line segment PQ equalsa)1b)√2c)√3d)2Correct answer is option 'C'. Can you explain this answer?.
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