Question 1316

Vectors II: Lines and Planes
2016 Paper 1 Question 11 Variant

Question

The plane p{p} has equation
r=(551)+λ(201)+μ(a35),\mathbf{r} = \begin{pmatrix} 5 \\ - 5 \\ - 1 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 0 \\ - 1 \end{pmatrix}+\mu\begin{pmatrix} a \\ 3 \\ 5 \end{pmatrix},
and the line l{l} has equation
r=(a+4a+6a+6)+t(122),\mathbf{r}=\begin{pmatrix} a + 4 \\ a + 6 \\ a + 6 \end{pmatrix}+t\begin{pmatrix} 1 \\ - 2 \\ 2 \end{pmatrix},
where a{a} is a constant and λ,μ{\lambda, \mu} and t{t} are parameters.
(i)
In the case where a=4,{a=- 4,}
(ia)
Show that l{l} is perpendicular to p{p} and find the values of λ,μ{\lambda, \mu} and t{t} which give the coordinates of the point at which l{l} and p{p} intersect,
[5]
(ib)
find the cartesian equations of the planes such that the perpendicular distance from each plane to p{p} is 8.{8.}
[5]
(ii)
Find the value of a{a} such that l{l} and p{p} do not meet in a unique point.
[3]

Answer