Let a line pass through two distinct points P ( − 2 , − 1 , 3 ) and Q , and be parallel to the vector 3 i ^ + 2 j ^ + 2 k ^ . If the distance of the point Q from the point R ( 1 , 3 , 3 ) is 5 , the

Question

Let a line pass through two distinct points P ( − 2 , − 1 , 3 ) and Q , and be parallel to the vector 3 i ^ + 2 j ^ ​ + 2 k ^ . If the distance of the point Q from the point R ( 1 , 3 , 3 ) is 5 , then the square of the area of △ PQR is equal to :

TestHub · Accepted Answer

Equation of line PQ is: 3 x + 2 ​ = 2 y + 1 ​ = 2 z − 3 ​ = r ( say ) Let coordinate of Q = ( 3 r − 2 , 2 r − 1 , 2 r + 3 ) ∵ PR = 5 Then ​ ( 3 r − 2 − 1 ) 2 + ( 2 r − 1 − 3 ) 2 + ( 2 r + 3 − 3 ) 2 = 25 ∴ r = 0 or 2 ∴ Coordinate of Q = ( 4 , 3 , 7 ) ​ ∴ ​ square of area of △ PQR = ​ 2 1 ​ ( PQ ​ × PR ) ​ 2 = ​ 2 1 ​ ( 6 i ^ + 4 j ^ ​ + 4 k ^ ) × ( 3 i ^ + 4 j ^ ​ ) ​ 2 = ∣ − 8 i ^ + 6 j ^ ​ + 6 k ^ ∣ 2 = 136 ​

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