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
5x14y12z=24,4x+4y+6z=12,5x+λy9z=μ, \begin{alignat*}{6} 5 x & \,-\, & 14 y & \,-\, & 12 z & = 24, \\ - 4 x & \,+\, & 4 y & \,+\, & 6 z & = - 12, \\ 5 x & \,+\, & \lambda y & \,-\, & 9 z & = \mu, \end{alignat*}
respectively, where λ{\lambda} and μ{\mu} are constants.
When λ=10.4{\lambda=-10.4} and μ=17.7,{\mu = 17.7,} 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 (4,5,2).{\left( 4, - 5, 2 \right).}
[4]

Answer