Let P ( α , β , γ ) be the image of the point Q ( 3 , − 3 , 1 ) in the line 1 x − 0 = 1 y − 3 = − 1 z − 1 and R be the point ( 2 , 5 , − 1 ) . If the area of the triangle PQR is λ and λ 2 = 14 K

Question

Let P ( α , β , γ ) be the image of the point Q ( 3 , − 3 , 1 ) in the line 1 x − 0 ​ = 1 y − 3 ​ = − 1 z − 1 ​ and R be the point ( 2 , 5 , − 1 ) . If the area of the triangle PQR is λ and λ 2 = 14 K , then K is equal to :

TestHub · Accepted Answer

​ RQ = 1 + 64 + 4 ​ = 69 ​ RQ ​ = ℓ ^ − 8 j ^ ​ + 2 k ^ RS = ℓ ^ + j ^ ​ − k ^ cos θ = ∣ RQ ​ ∣∣ RS ∣ RQ ​ ⋅ RS ​ = ​ 69 ​ 3 ​ 1 − 8 − 2 ​ ​ = 3 23 ​ 9 ​ cos θ = 23 ​ 3 ​ = RQ RS ​ = 69 ​ RS ​ RS = 3 3 ​ sin θ = 23 ​ 14 ​ ​ = 69 ​ QS ​ ​ ​ QS = 42 ​ area = 2 1 ​ ⋅ 2 QS ⋅ RS = 42 ​ ⋅ 3 3 ​ λ = 9 14 ​ λ 2 = 81.14 = 14 k k = 81 ​

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