2021 NYJC P2 Q1

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2021 NYJC P2 Q1

The plane ${{\prod }_{1}}$ passes through $\left( 3,-1,2 \right)$ and is perpendicular to the line $\mathbf{r}=2\mathbf{i}+4\mathbf{j}-3\mathbf{k}+\lambda \left( 5\mathbf{i}+2\mathbf{j}-\mathbf{k} \right)$. The plane ${{\prod }_{2}}$ contains the points $\left( 2,3,2 \right)$, $\left( 4,1,-1 \right)$ and $\left( 0,-1,2 \right)$.


Show that the acute angle, $\theta $, between the planes ${{\prod }_{1}}$ and ${{\prod }_{2}}$ is such that $\cos \theta =\frac{\sqrt{30}}{15}$.



Show that the line of intersection, $L$, of the planes ${{\prod }_{1}}$ and ${{\prod }_{2}}$ has vector equation $\mathbf{r}=\left( \begin{matrix}0 \\9 \\7 \\\end{matrix} \right)+\mu \left(\begin{matrix}-1 \\4 \\3 \\\end{matrix} \right)$, $\mu \in \mathbb{R}$.


The plane ${{\prod }_{3}}$ has the equation $4\left( k-2 \right)x+\left( k+1 \right)y-4{{k}^{2}}z=8$, where $k$ is a constant.


The three planes ${{\prod }_{1}}$,${{\prod }_{2}}$ and ${{\prod }_{3}}$have no points in common. By considering the relationship between the line $L$ and the plane ${{\prod }_{3}}$, find the possible values of $k$.



For the positive value of $k$ found in (iii), find the distance between $L$ and ${{\prod }_{3}}$.


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Published: 24th June 2022

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