Question 1308

Vectors II: Lines and Planes
2008 Paper 1 Question 11 Variant

Question

The equations of three planes p1,{p_1, } p2,{p_2, } p3{p_3 } are
6x+6z=12,4x+2y+6z=10,2x+λy+z=μ, \begin{alignat*}{6} 6 x & \,+\, & & \,-\, & 6 z & = - 12, \\ - 4 x & \,+\, & 2 y & \,+\, & 6 z & = 10, \\ - 2 x & \,+\, & \lambda y & \,+\, & z & = \mu, \end{alignat*}
respectively, where λ{\lambda} and μ{\mu} are constants.
When λ=3.8{\lambda=3.8} and μ=6,{\mu = 6,} find the coordinates of the point at which the three planes meet.
[2]
The planes p1{p_1} and p2{p_2} intersect in a line l.{l.}
(i)
Find a vector equation of l.{l.}
[4]
(ii)
Given that l{l} lies in p3,{p_3,} find λ{\lambda} and μ.{\mu.}
[3]
(iii)
Given instead that l{l} and p3{p_3} have no point in common, what can be said about the values of λ{\lambda} and μ.{\mu.}
[2]
(iv)
Find the cartesian equation of the plane which contains l{l} and the point (5,4,2).{\left( - 5, 4, - 2 \right).}
[4]

Answer