If the image of the point (4,4,3) in the line 2x−1=1y−2=3z−1 is (α,β,γ), then α+β+γ is equal to
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Step-by-Step Solution
Step 1: Find a general point on the line
We start by setting the given equation of the line equal to a parameter λ. This allows us to express the coordinates of any point on the line in terms of λ.
Step 2: Determine the foot of the perpendicular
Let A(4,4,3) be the given point and P(1+2λ,2+λ,1+3λ) be a general point on the line. The vector AP connects point A to point P. Since P is the foot of the perpendicular from A to the line, AP must be perpendicular to the direction vector of the line, d=(2,1,3). Their dot product must be zero.
Step 3: Solve for λ
We expand the dot product equation and solve for λ. This value of λ will give us the coordinates of the foot of the perpendicular from point A to the line.
Step 4: Find the coordinates of the foot of the perpendicular
Substitute λ=1 back into the general point P(1+2λ,2+λ,1+3λ) to find the coordinates of the foot of the perpendicular, P.
Step 5: Find the image point
Let the image of point A(4,4,3) be A′(α,β,γ). The foot of the perpendicular P is the midpoint of the segment AA′. We use the midpoint formula to find the coordinates of A′.
Step 6: Calculate the sum of the coordinates of the image point
Finally, we sum the coordinates of the image point (α,β,γ) to get the required value.