2022HSCShort answer

Q11 (e)

2 marks

Question

Let $\ell_1$ be the line with equation
$\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}-1\\7\end{pmatrix}+\lambda\begin{pmatrix}3\\2\end{pmatrix},\quad \lambda\in\mathbb R.$

The line $\ell_2$ passes through the point $A(-6,5)$ and is parallel to $\ell_1$ . Find the equation of the line $\ell_2$ in the form $y=mx+c$ .

Try the question before revealing the answer.

Solution

Criteria	Marks
Provides correct solution	2
Finds equation of $\ell_2$ in any form OR finds the correct slope of $\ell_2$ OR equivalent merit	1

Sample answer:

\ell_1:\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}-1\\7\end{pmatrix}+\lambda\begin{pmatrix}3\\2\end{pmatrix}

$\ell_2\parallel\ell_1$ passes through $(-6,5)$ .

\ell_2:\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}-6\\5\end{pmatrix}+\mu\begin{pmatrix}3\\2\end{pmatrix}

$x=-6+3\mu,\quad y=5+2\mu$
$2x=-12+6\mu,\quad 3y=15+6\mu$
Subtract:
$3y-2x=27$
$3y=2x+27$
$y=\frac23x+9$

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Syllabus alignment

Vectors

Vectors and vector equations of linesUse coordinates and vector equations for curves, lines, line segments and spheres in two and three dimensions.

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