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Question
Find the equations of the line passing through the point (−1, 2, 1) and parallel to the line \[\frac{2x - 1}{4} = \frac{3y + 5}{2} = \frac{2 - z}{3} .\]
Solution
The equation of line \[\frac{2x - 1}{4} = \frac{3y + 5}{2} = \frac{2 - z}{3}\] can be re-written as \[\frac{x - \frac{1}{2}}{2} = \frac{y + \frac{5}{3}}{\frac{2}{3}} = \frac{z - 2}{- 3}\]
The direction ratios of the line parallel to line \[\frac{2x - 1}{4} = \frac{3y + 5}{2} = \frac{2 - z}{3}\] are proportional to 2, \[\frac{2}{3}\] \[-3\] . Equation of the required line passing through the point ( -1, 2, 1) having direction ratios proportional to 2, \[\frac{2}{3}\] \[-3\] is
\[\frac{x - \left( - 1 \right)}{2} = \frac{y - 2}{\frac{2}{3}} = \frac{z - 1}{- 3}\]
\[ = \frac{x + 1}{2} = \frac{y - 2}{\frac{2}{3}} = \frac{z - 1}{- 3}\]
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