Let P=(a,b,0) and the midpoint of PA be M=(0,43,−41).
Using the midpoint formula, the coordinates of A are given by 2M−P:
A=(−a,23−b,−21)
Since A lies on the given line 2x−1=1y−2=3z−α=λ, we can write:
2−a−1=λ⇒a=−2λ−1
123−b−2=λ⇒b=−λ−21
3−21−α=λ⇒α=−3λ−21
The direction ratios of the line segment PA are the differences in coordinates of A and P:
PA=(−2a,23−2b,−21)
Since PA is perpendicular to the given line, the dot product of their direction ratios must be zero. The direction ratios of the line are (2,1,3).
2(−2a)+1(23−2b)+3(−21)=0
−4a+23−2b−23=0
−4a−2b=0⇒2a+b=0
Substituting the expressions for a and b in terms of λ:
2(−2λ−1)+(−λ−21)=0
−4λ−2−λ−21=0
−5λ−25=0⇒λ=−21
Now, substituting λ=−21 back to find a,b, and α:
a=−2(−21)−1=0
b=−(−21)−21=0
α=−3(−21)−21=23−21=1
Finally, calculating the required value:
a2+b2+α2=02+02+12=1
Answer: 1