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What is the Cartesian equation of the line \( r = \left( \frac{1}{3} \right) + \lambda \left( -\frac{2}{4} \right) ? - HSC - SSCE Mathematics Extension 2 - Question 3 - 2020 - Paper 1
Question 3
What is the Cartesian equation of the line \( r = \left( \frac{1}{3} \right) + \lambda \left( -\frac{2}{4} \right) ?
Worked Solution & Example Answer:What is the Cartesian equation of the line \( r = \left( \frac{1}{3} \right) + \lambda \left( -\frac{2}{4} \right) ? - HSC - SSCE Mathematics Extension 2 - Question 3 - 2020 - Paper 1
Step 1
Determine the direction vector
Answer
The line is given in the form ( r = \mathbf{a} + \lambda \mathbf{b} ), where ( \mathbf{a} ) is the point ( \left( \frac{1}{3}, 0 \right) ) and ( \mathbf{b} ) is the direction vector ( \left( -\frac{2}{4} \right) ). Simplifying the direction vector gives ( \left( -\frac{1}{2}, 1 \right) ).
Step 2
Find Cartesian coordinates
Answer
To convert to Cartesian coordinates, we can express the components as follows:
[ x = \frac{1}{3} - \frac{1}{2} \lambda ] [ y = 0 + 1 \lambda ] Thus, substituting ( \lambda ) from the first equation into the second gives: [ \lambda = y ] [ x = \frac{1}{3} - \frac{1}{2} y ]
Step 3
Rearranging to find the equation
Answer
Rearranging the expression for ( x ): [ \frac{1}{2} y = \frac{1}{3} - x ] Multiplying through by 2 results in: [ y = -2x + 1/3 ] To clear fractions, eliminate the denominator: [ 2y - 4x = 2/3 ] In simplified form, we identify the correct choice from the options.
Step 4