The two planes ${{p}_{1}}$ and ${{p}_{2}}$, given by the equations $2x+y=1$ and $8x+ay+z=4$ respecti

Question

The two planes ${{p}_{1}}$ and ${{p}_{2}}$, given by the equations $2x+y=1$ and $8x+ay+z=4$ respectively, meet at a line $L$ which contains the point $A\left( 0,1,0 ight)$. Show that $a=4$ and hence find the vector equation of the line $L$. [3] Another point $B$ lies in the plane ${{p}_{1}}$ such that $AB$ is perpendicular to the line $L$. Show that $\overrightarrow{AB}$ is parallel to $\left( \begin{matrix} 0 \ 0 \ 1 \ \end{matrix} ight)$. If the distance of $B$ from ${{p}_{2}}$ is 5 units, find the possible position vectors of point $B$. [5] Find the acute angle between line $AB$ and ${{p}_{2}}$. [2]

Accepted Answer

(i) $a = 4$; $L: \mathbf{r} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} + \lambda\begin{pmatrix} -1 \\ -1 \\ 0 \end{pmatrix}$ (ii) $\overrightarrow{AB}$ is parallel to $\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$; $\overrightarrow{OB} = \begin{pmatrix} 0 \\ 1 \\ -45 \end{pmatrix}$ or $\begin{pmatrix} 0 \\ 1 \\ 45 \end{pmatrix}$ (iii) $6.4°$

2012 AJC P1 Q10

2012 AJC P1 Q10

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