Consider a plane P : 2 x + y - z = 5 , a line L : x - 3 2 = y + 1 - 3 = z - 2 - 1 and a point A 3 , - 4,1 . If the line L intersects plane P at B and the x y plane at C , then the area (in sq. units) of ∆ A B C is

Question

Consider a plane P : 2 x + y - z = 5 , a line L : x - 3 2 = y + 1 - 3 = z - 2 - 1 and a point A 3 , - 4,1 . If the line L intersects plane P at B and the x y plane at C , then the area (in sq. units) of ∆ A B C is, 7, 8, 10, 2 3

MARKS App · Accepted Answer

For x y plane, putting z = 0 , we get, x - 3 2 = y + 1 - 3 = 2 ⇒ x = 7 , y = - 7 Coordinates of point C are 7 , - 7,0 Any general point on the line is 2 λ + 3 , - 3 λ - 1 , - λ + 2 This must satisfy the given plane 4 λ + 6 - 3 λ - 1 + λ - 2 = 5 2 λ + 3 = 5 ⇒ λ = 1 So, coordinates of point B are 5 , - 4,1 A B → = 2,0 , 0 > A C → = 4 , - 3 , - 1 > Area = 1 2 A B → × A C → = 1 2 i ^ j ^ k ^ 2 0 0 4 - 3 - 1 = 1 2 - 2 - j ^ + 3 k ^ = j ^ - 3 k ^ = 10 sq. units

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