What is the Cartesian equation of the line \( \mathbf{r} = \left( \frac{1}{3} \right) + \lambda \left( -\frac{2}{4} \right) ?
A - HSC - SSCE Mathematics Extension 2 - Question 3 - 2020 - Paper 1
Question 3
What is the Cartesian equation of the line \( \mathbf{r} = \left( \frac{1}{3} \right) + \lambda \left( -\frac{2}{4} \right) ?
A. 2y + x = 7
B. y - 2x = -5
C. y + 2x ... show full transcript
Worked Solution & Example Answer:What is the Cartesian equation of the line \( \mathbf{r} = \left( \frac{1}{3} \right) + \lambda \left( -\frac{2}{4} \right) ?
A - HSC - SSCE Mathematics Extension 2 - Question 3 - 2020 - Paper 1
Step 1
Identify the direction vector
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Answer
From the given line equation ( \mathbf{r} = \left( \frac{1}{3}, y \right) + \lambda \left( -\frac{2}{4} \right) ), we identify the direction vector as ( \mathbf{d} = \begin{pmatrix} -2 \ 4 \end{pmatrix} ). The equivalent vector is given as ( \mathbf{d} = \begin{pmatrix} -2 \ 2 \end{pmatrix} ) when simplified.
Step 2
Determine the slope
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Answer
To find the slope of the line, we can express it in terms of ( y ) as follows:
The slope ( m ) can be calculated as ( m = \frac{\text{rise}}{\text{run}} = \frac{2}{-2} = -1 ).
Step 3
Find the y-intercept
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Answer
To derive the equation from the slope, we can use the point ( \left( \frac{1}{3}, y_0 \right) ). Setting ( y = mx + b ) and using the point-to-form, we can find ( b = y_0 - mx_0 ). We substitute and simplify to determine the y-intercept.
Step 4
Construct the Cartesian equation
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Answer
Using the slope-intercept form ( y = mx + b ), we convert to standard form. We solve for ( y + 2x = 5 ) which can be arranged to find the correct equation. Thus, the Cartesian equation of the line is ( y + 2x = 5 ). This matches option C.
