A plane P is parallel to two lines whose direction ratios are - 2 , 1 , - 3 , and - 1 , 2 , - 2 and it contains the point 2 , 2 , - 2 . Let P intersect the co-ordinate axes at the points A , B , C making the intercepts α , β , γ . If V is the volume of the tetrahedron O A B C , where O is the origin and p = α + β + γ , then the ordered pair V , p is equal to

Step-by-step solution

Given a plane P is parallel to two lines whose direction ratios are - 2 , 1 , - 3 , and - 1 , 2 , - 2 , So normal of plane P is given by, i ^ j ^ k ^ - 2 1 - 3 - 1 2 - 2 = 4 i ^ - j ^ - 3 k ^ Now equation of plane P which passes through 2 , 2 , - 2 will be, 4 x - 2 - y - 2 - 3 z + 2 = 0 ⇒ 4 x - y - 3 z - 12 = 0 Now, coordinates of plane making intercepts at X , Y   &   Z axis will be A 3 , 0 , 0 , B 0 , - 12 , 0 , C 0 , 0 , - 4 ⇒ α = 3 , β = - 12 , γ = - 4 ⇒ p = α