If the image of the point 1 , - 2 , 3 in the plane 2 x + 3 y - z = 7 is the point α , β , γ , then the value of α + β + γ is equal to

Step-by-step solution

Since, α , β , γ is the image of the point 1 , - 2 , 3 in the plane. So, we have, ∵ Image of ( x 1 , y 1 , z 1 ) in the plane a x + b y + c z + d = 0 is given by x − x 1 a = y − y 1 b = z − z 1 c = − 2 ( a x 1 + b y 1 + c z 1 ) a 2 + b 2 + c 2 α - 1 2 = β + 2 3 = γ - 3 - 1 = - 2 2 1 + 3 - 2 - 3 - 7 2 2 + 3 2 + - 1 2 ⇒ α - 1 2 = β + 2 3 = γ - 3 - 1 = - 2 2 - 6 - 3 - 7 4 + 9 + 1 ⇒ α - 1 2 = β + 2 3 = γ - 3 - 1 = - 2 ∴ α - 1 2 = - 2 ⇒ α = - 4 + 1 = - 3 β + 2 3 = - 2 ⇒ β = - 6 - 2 = - 8 and γ - 3 - 1 = - 2 ⇒ γ = 2 + 3