2022 ACJC Promo Q10
The plane ${{P}_{1}}$ has the equation $-6x-4y+2z=4$.
(i)
Find the vector equations of the planes such that the perpendicular distance from each plane to ${{P}_{1}}$ is $10$ units.
[2]
The plane ${{P}_{2}}$ has the equation $-x-y+2z=k$, where $k$ is a constant.
(ii)
Find the angle between ${{P}_{1}}$ and ${{P}_{2}}$.
[2]
(iii)
The planes ${{P}_{1}}$ and ${{P}_{2}}$ intersect in the line $L$. Show that a possible vector equation of $L$ is $\mathbf{r}=\left( \begin{matrix}
2k-2 \\
2-3k \\
0 \\
\end{matrix} \right)+\lambda \left( \begin{matrix}
-3 \\
5 \\
1 \\
\end{matrix} \right)$, $\lambda \in \mathbb{R}$.
[3]
The plane ${{P}_{3}}$ has the equation $5x+\beta y+5z=\mu $, where $\beta $, $\mu \in \mathbb{R}$.
(iv)
Given that the line $L$ is contained in the plane ${{P}_{3}}$, find $\beta $ and $\mu $, giving your answer in terms of $k$ if necessary.
[2]
(v)
Given instead that the line $L$ does not intersect ${{P}_{3}}$, what can be said about the values of $\beta $ and $\mu $?
[1]
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