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+5y+z=4,x+3y2z=8,5x+λy18z=μ, \begin{alignat*}{6} - 6 x & \,+\, & 5 y & \,+\, & z & = 4, \\ - x & \,+\, & 3 y & \,-\, & 2 z & = - 8, \\ - 5 x & \,+\, & \lambda y & \,-\, & 18 z & = \mu, \end{alignat*}
respectively, where λ{\lambda} and μ{\mu} are constants.
When λ=18{\lambda=18} and μ=66.5,{\mu = -66.5,} 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 (2,4,2).{\left( 2, 4, 2 \right).}
[4]

Answer