The line r=(2,3,4)+µ(-1,0,3) and the point (6,0,2) form which plane: (a) r.(-1,0,3)=10 (b) r.(9,10,3)=60 (c) r.(2,3,4)=29 (d) r=(2,3,4)+µ(-1,0,3)+ù(6,0,2)
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To determine which plane is formed by the line and the given point, we need to substitute the coordinates of the line and the point into the equations of the planes and see which equation is satisfied. Let's go through each option:
(a) r.(-1,0,3) = 10
First, we substitute the coordinates of the line: (2,3,4) + µ(-1,0,3) it becomes (2-µ,3,4+3µ).
Then, we substitute the coordinates of the point (6,0,2) into the equation: (2-µ)(-1) + 3µ(0) + (4+3µ)(3) = 10.
Simplifying the equation, we get -2 + 6µ + 12 + 9µ = 10.
Combine like terms: 15µ + 10 = 10.
Since the equation becomes an identity and is always true regardless of the value of µ, this plane is satisfied by the line and the point.
(b) r.(9,10,3) = 60
Substituting the coordinates of the line into the equation: (2-µ)(9) + 3µ(10) + (4+3µ)(3) = 60.
Simplifying, we get 18 - 9µ + 30µ + 12 + 9µ = 60.
Combine like terms: 30µ + 30 = 60.
Solving for µ, we have µ = 1.
However, this value of µ does not satisfy the equation for the given point since the left side evaluates to 60, but the right side is 60.
Therefore, this plane is not satisfied by the line and the point.
(c) r.(2,3,4) = 29
Substituting the coordinates of the line into the equation: (2-µ)(2) + 3µ(3) + (4+3µ)(4) = 29.
Simplifying the equation, we get 4 - 2µ + 9µ + 16 + 12µ = 29.
Combine like terms: 19µ + 20 = 29.
Solving for µ, we have µ = 0.47 (rounded to two decimal places).
However, this value of µ does not satisfy the equation for the given point since the left side evaluates to 29, but the right side is 29.
Therefore, this plane is not satisfied by the line and the point.
(d) r = (2,3,4) + µ(-1,0,3) + ù(6,0,2)
This option is not an equation of a plane but is the parametric equation of the line passing through the point (2,3,4) parallel to the vector (-1,0,3).
Therefore, this does not represent a plane.
Based on our analysis, the correct answer is (a) r.(-1,0,3)=10. This plane is satisfied by the given line and point.