Consider a line L passing through the points P ( 1 , 2 , 1 ) and Q ( 2 , 1 , − 1 ) . If the mirror image of the point A ( 2 , 2 , 2 ) in the line L is ( α , β , γ ) , then α + β + 6 γ is equal to ____

Question

Consider a line L passing through the points P ( 1 , 2 , 1 ) and Q ( 2 , 1 , − 1 ) . If the mirror image of the point A ( 2 , 2 , 2 ) in the line L is ( α , β , γ ) , then α + β + 6 γ is equal to _______

TestHub · Accepted Answer

DR's of Line L ≡ − 1 : 1 : 2 DR's of AB ≡ α − 2 : β − 2 : γ − 2 ​ AB ⊥ ar ​ L ⇒ 2 − α + β − 2 + 2 γ − 4 = 0 2 γ + β − α = 4 ​ Let C is mid-point of A B ​ C ( 2 α + 2 ​ , 2 β + 2 ​ , 2 γ + 2 ​ ) DR’s of PC = 2 α ​ : 2 β − 2 ​ : 2 γ ​ line L ​ PC ⇒ 2 − α ​ = 2 β − 2 ​ = 4 γ ​ = K ( let) ​ ​ α = − 2 K β = 2 K + 2 γ = 4 K ​ use in (1) ⇒ K = 6 1 ​ value of α + β + 6 γ = 24 K + 2 = 6

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