2023 EJC P2 Q5
2023 EJC P2 Q5
11 marks
Prelims
Planes \(p\) and \(q\) are perpendicular. Plane \(p\) has equation \(2x-z=10\). Plane \(q\) contains the line \(l\) with equation
\(\mathbf{r}=\left( \begin{matrix}
\begin{matrix}
5 \\
-3 \\
4 \\
\end{matrix} \\
\end{matrix} \right)+\lambda \left( \begin{matrix}
\begin{matrix}
2 \\
-2 \\
1 \\
\end{matrix} \\
\end{matrix} \right)\), where \(\lambda \) is a parameter.
- Show that \(q\) has equation \(x+2y+2z=7\).[2]
- Find a vector equation of the line \(m\) where \(p\) and \(q\) meet.[2]
The point \(A\) with coordinates \(\left( 2,\,1,\,-6 \right)\) lies on \(p\) and the shortest distance from \(A\) to \(q\) is \(k\).
- Find the position vector of the foot of perpendicular, \(F\) from \(A\) to \(q\). Hence find the value of \(k\).[4]
- Find vector equations of the lines in \(p\) such that the shortest distance from each line to \(q\) is \(k\).[3]
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Answer:(a) \(q:x+2y+2z=7\) (b) \(\mathbf r=\begin{pmatrix}5\\1\\0\end{pmatrix}+\mu\begin{pmatrix}2\\-5\\4\end{pmatrix}\), \(\mu\in\mathbb R\) (c) \(\overrightarrow{OF}=\begin{pmatrix}11/3\\13/3\\-8/3\end{pmatrix}\), \(k=5\) (d) \(\mathbf r=\begin{pmatrix}2\\1\\-6\end{pmatrix}+\alpha\begin{pmatrix}2\\-5\\4\end{pmatrix}\) and \(\mathbf r=\begin{pmatrix}16/3\\23/3\\2/3\end{pmatrix}+\beta\begin{pmatrix}2\\-5\\4\end{pmatrix}\), \(\alpha,\beta\in\mathbb R\)
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