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) $t = -\frac{22}{3}$ or $t = 2$ (b)(ii) $\overrightarrow{OA} = \begin{pmatrix} -2/9 \ -2/3 \ 37/9 \end{pmatrix}, \overrightarrow{OB} = \begin{pmatrix} 14/9 \ -22/9 \ 5 \end{pmatrix}$ (iii) $\mathbf{r} = \begin{pmatrix} 10/3 \ -38/9 \ 53/9 \end{pmatrix} + \beta \begin{pmatrix} 4 \ 3 \ -2 \end{pmatrix}; \beta \in \mathbb{R}$

2013 ACJC P1 Q13

2013 ACJC P1 Q13

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