Let S be the reflection of a point Q with respect to the plane given by r → = - t + p i ^ + t j ^ + 1 + p k ^ where t , p are real parameters and i ^ , j ^ , k ^ are the unit vectors along the three positive coordinate axes. If the position vectors of Q and S are 10 i ^ + 15 j ^ + 20 k ^ and α i ^ + β j ^ + γ k ^ respectively, then which of the following is/are TRUE?
C. 3 γ + α = - 86
Given, Equation of plane is r → = - t + p i ^ + t j ^ + 1 + p k ^ On rearranging we get, r → = k ^ + t - i ^ + j ^ + p - i ^ + k ^ So, equation of plane in standard form is given by r → - k ^ - i ^ + j ^ - i ^ + k ^ = 0 ⇒ x y z - 1 - 1 1 0 - 1 0 1 = 0 ⇒ x + y + z = 1             ⋯ 1 Now given coordinate of Q = 10 , 15 , 20 And coordinates of S = α , β , γ Now using the image formula of point and plane we get, α - 10 1 = β - 15 1 = γ - 20 1 = - 2 10 + 15 + 20 - 1 3 ⇒ α - 10 = β - 15 = γ - 20 = - 88 3 ⇒ α = - 58 3 , β = - 43 3 , γ = - 28 3 Now solving all options we get, 3 α + β = - 101 3 β + γ = - 71 3 γ + α = - 86 3 α + β + γ = - 129
Related: Mathematics — Three Dimensional Geometry · All PYQ Banks