The line ${{l}_{1}}$ passes through the point $A\left( 7,6,5 \right)$ and the point \(B\left( -5

Question

The line ${{l}_{1}}$ passes through the point $A\left( 7,6,5 \right)$ and the point $B\left( -5,-2,-5 \right)$ while the line ${{l}_{2}}$ has equation $x-5=\frac{y-10}{k}=\frac{z-8}{2}$, where $k$ is a constant. The lines ${{l}_{1}}$ and ${{l}_{2}}$ intersect at point $P$. Show that $k=2$ and find the coordinates of $P$.[5] Find the position vector of the point on ${{l}_{2}}$ which is closest to $A$.[3] Hence, find a vector equation of the line which is a reflection of ${{l}_{1}}$ in ${{l}_{2}}$.[2]

Accepted Answer

(i) $k=2$, $P=(1,2,0)$ (ii) $\overrightarrow{OF}=\begin{pmatrix}\frac{11}{3}\\frac{22}{3}\\frac{16}{3}\end{pmatrix}$ (iii) $\mathbf r=\begin{pmatrix}1\2\0\end{pmatrix}+t\begin{pmatrix}-2\20\17\end{pmatrix}$, where $t\in\mathbb R$

2024 NJC Promo Q5

2024 NJC Promo Q5

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