The equations of the plane $\pi$, and the lines ${{l}_{1}}$ and ${{l}_{2}}$ are given by \(\be

Question

The equations of the plane $\pi$, and the lines ${{l}_{1}}$ and ${{l}_{2}}$ are given by $\begin{aligned} & \pi :tx-2y+z=13 \ & {{l}_{1}}:\mathbf{r}=\left( 2\mathbf{i}+\mathbf{j}+3\mathbf{k} \right)+\lambda \left[ 2t\mathrm{ }\mathbf{i}+\left( {{t}^{2}}-1 \right)\mathbf{j}-2\mathbf{k} \right] \ & {{l}_{2}}:x-4=y,z=5, \ \end{aligned}$ where $t$ and $\lambda $ are real constants. Given that the shortest distance from the point P with coordinates $\left( 3,3,-5 \right)$ to $\pi$ is $6$, find the possible values of $t$.[4] Show that ${{l}_{1}}$ is parallel to $\pi$, and find a condition of $t$ such that ${{l}_{1}}$ is not on $\pi$. For the case where $t=2$,[3] show that ${{l}_{2}}$ lies on $\pi$,[1] given that $A$ is a point on ${{l}_{1}}$ and $B$ is a point on ${{l}_{2}}$, find the position vectors of $A$ and $B$ such that $AB$ is perpendicular to both ${{l}_{1}}$ and ${{l}_{2}}$,[4] find the vector equation of the line of reflection of ${{l}_{1}}$ in $\pi$.[2]

Accepted Answer

(a) $t=2$ or $t=-\frac{22}{3}$. (b) $l_1\parallel\pi$, and $l_1$ is not on $\pi$ when $t\ne6$. For $t=2$: (i) $l_2$ lies on $\pi$; (ii) $\overrightarrow{OA}=\begin{pmatrix}-\frac29 \ -\frac23 \ \frac{37}{9}\end{pmatrix}$, $\overrightarrow{OB}=\begin{pmatrix}\frac{14}{9} \ -\frac{22}{9} \ 5\end{pmatrix}$; (iii) $\mathbf r=\begin{pmatrix}\frac{10}{3} \ -\frac{38}{9} \ \frac{53}{9}\end{pmatrix}+\beta\begin{pmatrix}4 \ 3 \ -2\end{pmatrix}$, $\beta\in\mathbb R$.

2013 ACJC P1 Q13

2013 ACJC P1 Q13

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