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Let P be a point in the first octant, whose image Q in the plane x + y= 3 (that is, the line segment PQ is perpendicular to the plane x + y= 3 and the mid-point of PQ lies in the plane x + y= 3) lies on the z-axis. Let the distance of P from the x- axis be 5. If R is the image of P in the xy-plane, then the length of PR is ____.
Correct answer is '8'. Can you explain this answer?
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Let P be a point in the first octant, whose image Q in the plane x + y...
The coordinates for point P and point Q are given as,
The equation of image of Q on the plane is given as,
The coordinates of point Q are (3 − β ,3−α ,γ) .
Since, point Q lies on the z - axis. Therefore,
The coordinates of point P are (3, 3,γ) and the distance of point P from x- axis is 5. Therefore, find the value of γ.
The coordinates of point P are (3, 3, 4) and R are (3, 3, −4) .
Calculate the distance between points P and R.
The equation of image of Q on the plane is given as,
The coordinates of point Q are (3 − β ,3−α ,γ) .
Since, point Q lies on the z - axis. Therefore,
The coordinates of point P are (3, 3,γ) and the distance of point P from x- axis is 5. Therefore, find the value of γ.
The coordinates of point P are (3, 3, 4) and R are (3, 3, −4) .
Calculate the distance between points P and R.
Most Upvoted Answer
Let P be a point in the first octant, whose image Q in the plane x + y...
Given information:
- Point P is in the first octant.
- Image Q of point P lies on the plane x + y = 3.
- Point Q is also on the z-axis.
- Distance of point P from the x-axis is 5.
- Image R of point P lies in the xy-plane.
Understanding the problem:
We need to find the length of PR, the distance between the image R of point P and point P itself. We are given certain conditions about the points P, Q, and R and their positions in space.
Solution:
Step 1: Finding the coordinates of point P
Since point P is in the first octant and its distance from the x-axis is 5, we can determine its coordinates as (5, y, z), where y and z are the y-coordinate and z-coordinate of point P, respectively.
Step 2: Finding the coordinates of point Q
The image Q of point P lies on the plane x + y = 3 and also on the z-axis. Therefore, the x-coordinate of point Q is 0, the y-coordinate is 3, and the z-coordinate is 0.
Step 3: Finding the coordinates of point R
The image R of point P lies in the xy-plane. Since the z-coordinate of point P is z, the z-coordinate of point R will be the negative of z, i.e., -z. Therefore, the coordinates of point R are (5, y, -z).
Step 4: Finding the distance between point P and point R
The distance between two points in 3D space can be found using the distance formula. Considering points P and R, the formula becomes:
PR = √[(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2]
Substituting the coordinates of points P and R, we get:
PR = √[(5 - 5)^2 + (y - y)^2 + (z - (-z))^2]
PR = √[0 + 0 + 4z^2]
PR = 2z
Step 5: Finding the value of z
We know that the image Q of point P lies on the z-axis, so its x-coordinate and y-coordinate are both 0. Therefore, the coordinates of point Q are (0, 3, 0). Since point Q is the midpoint of line segment PQ, the coordinates of point P can be found as follows:
x-coordinate of P = 2 * 0 - 5 = -5
y-coordinate of P = 2 * 3 - y = 6 - y
z-coordinate of P = 2 * 0 - z = -z
The coordinates of point P are (-5, 6 - y, -z).
Step 6: Using the given conditions to find the value of y
Point P lies on the plane x + y = 3. Substituting the x and y coordinates of point P, we get:
-5 + (6 - y) = 3
1 - y = 3
y = -2
Therefore, the value of y is -2.
Step
- Point P is in the first octant.
- Image Q of point P lies on the plane x + y = 3.
- Point Q is also on the z-axis.
- Distance of point P from the x-axis is 5.
- Image R of point P lies in the xy-plane.
Understanding the problem:
We need to find the length of PR, the distance between the image R of point P and point P itself. We are given certain conditions about the points P, Q, and R and their positions in space.
Solution:
Step 1: Finding the coordinates of point P
Since point P is in the first octant and its distance from the x-axis is 5, we can determine its coordinates as (5, y, z), where y and z are the y-coordinate and z-coordinate of point P, respectively.
Step 2: Finding the coordinates of point Q
The image Q of point P lies on the plane x + y = 3 and also on the z-axis. Therefore, the x-coordinate of point Q is 0, the y-coordinate is 3, and the z-coordinate is 0.
Step 3: Finding the coordinates of point R
The image R of point P lies in the xy-plane. Since the z-coordinate of point P is z, the z-coordinate of point R will be the negative of z, i.e., -z. Therefore, the coordinates of point R are (5, y, -z).
Step 4: Finding the distance between point P and point R
The distance between two points in 3D space can be found using the distance formula. Considering points P and R, the formula becomes:
PR = √[(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2]
Substituting the coordinates of points P and R, we get:
PR = √[(5 - 5)^2 + (y - y)^2 + (z - (-z))^2]
PR = √[0 + 0 + 4z^2]
PR = 2z
Step 5: Finding the value of z
We know that the image Q of point P lies on the z-axis, so its x-coordinate and y-coordinate are both 0. Therefore, the coordinates of point Q are (0, 3, 0). Since point Q is the midpoint of line segment PQ, the coordinates of point P can be found as follows:
x-coordinate of P = 2 * 0 - 5 = -5
y-coordinate of P = 2 * 3 - y = 6 - y
z-coordinate of P = 2 * 0 - z = -z
The coordinates of point P are (-5, 6 - y, -z).
Step 6: Using the given conditions to find the value of y
Point P lies on the plane x + y = 3. Substituting the x and y coordinates of point P, we get:
-5 + (6 - y) = 3
1 - y = 3
y = -2
Therefore, the value of y is -2.
Step
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Let P be a point in the first octant, whose image Q in the plane x + y= 3 (that is, the line segment PQ is perpendicular to the plane x + y= 3 and the mid-point of PQ lies in the plane x + y= 3)lies on the z-axis. Let the distance of P from the x-axis be 5. If R is the image of P in the xy-plane, then the length of PR is ____.Correct answer is '8'. Can you explain this answer?
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Let P be a point in the first octant, whose image Q in the plane x + y= 3 (that is, the line segment PQ is perpendicular to the plane x + y= 3 and the mid-point of PQ lies in the plane x + y= 3)lies on the z-axis. Let the distance of P from the x-axis be 5. If R is the image of P in the xy-plane, then the length of PR is ____.Correct answer is '8'. 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 Let P be a point in the first octant, whose image Q in the plane x + y= 3 (that is, the line segment PQ is perpendicular to the plane x + y= 3 and the mid-point of PQ lies in the plane x + y= 3)lies on the z-axis. Let the distance of P from the x-axis be 5. If R is the image of P in the xy-plane, then the length of PR is ____.Correct answer is '8'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let P be a point in the first octant, whose image Q in the plane x + y= 3 (that is, the line segment PQ is perpendicular to the plane x + y= 3 and the mid-point of PQ lies in the plane x + y= 3)lies on the z-axis. Let the distance of P from the x-axis be 5. If R is the image of P in the xy-plane, then the length of PR is ____.Correct answer is '8'. Can you explain this answer?.
Let P be a point in the first octant, whose image Q in the plane x + y= 3 (that is, the line segment PQ is perpendicular to the plane x + y= 3 and the mid-point of PQ lies in the plane x + y= 3)lies on the z-axis. Let the distance of P from the x-axis be 5. If R is the image of P in the xy-plane, then the length of PR is ____.Correct answer is '8'. 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 Let P be a point in the first octant, whose image Q in the plane x + y= 3 (that is, the line segment PQ is perpendicular to the plane x + y= 3 and the mid-point of PQ lies in the plane x + y= 3)lies on the z-axis. Let the distance of P from the x-axis be 5. If R is the image of P in the xy-plane, then the length of PR is ____.Correct answer is '8'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let P be a point in the first octant, whose image Q in the plane x + y= 3 (that is, the line segment PQ is perpendicular to the plane x + y= 3 and the mid-point of PQ lies in the plane x + y= 3)lies on the z-axis. Let the distance of P from the x-axis be 5. If R is the image of P in the xy-plane, then the length of PR is ____.Correct answer is '8'. Can you explain this answer?.
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Let P be a point in the first octant, whose image Q in the plane x + y= 3 (that is, the line segment PQ is perpendicular to the plane x + y= 3 and the mid-point of PQ lies in the plane x + y= 3)lies on the z-axis. Let the distance of P from the x-axis be 5. If R is the image of P in the xy-plane, then the length of PR is ____.Correct answer is '8'. Can you explain this answer?, a detailed solution for Let P be a point in the first octant, whose image Q in the plane x + y= 3 (that is, the line segment PQ is perpendicular to the plane x + y= 3 and the mid-point of PQ lies in the plane x + y= 3)lies on the z-axis. Let the distance of P from the x-axis be 5. If R is the image of P in the xy-plane, then the length of PR is ____.Correct answer is '8'. Can you explain this answer? has been provided alongside types of Let P be a point in the first octant, whose image Q in the plane x + y= 3 (that is, the line segment PQ is perpendicular to the plane x + y= 3 and the mid-point of PQ lies in the plane x + y= 3)lies on the z-axis. Let the distance of P from the x-axis be 5. If R is the image of P in the xy-plane, then the length of PR is ____.Correct answer is '8'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Let P be a point in the first octant, whose image Q in the plane x + y= 3 (that is, the line segment PQ is perpendicular to the plane x + y= 3 and the mid-point of PQ lies in the plane x + y= 3)lies on the z-axis. Let the distance of P from the x-axis be 5. If R is the image of P in the xy-plane, then the length of PR is ____.Correct answer is '8'. Can you explain this answer? tests, examples and also practice JEE tests.
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