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Question
Find the length of the perpendicular from the point (4, −7) to the line joining the origin and the point of intersection of the lines 2x − 3y + 14 = 0 and 5x + 4y − 7 = 0.
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Solution
Solving the lines 2x − 3y + 14 = 0 and 5x + 4y − 7 = 0 we get:
\[\frac{x}{21 - 56} = \frac{y}{70 + 14} = \frac{1}{8 + 15}\]
\[ \Rightarrow x = - \frac{35}{23}, y = \frac{84}{23}\]
So, the point of intersection of 2x − 3y + 14 = 0 and 5x + 4y − 7 = 0 is \[\left( - \frac{35}{23}, \frac{84}{23} \right)\] .
The equation of the line passing through the origin and the point \[\left( - \frac{35}{23}, \frac{84}{23} \right)\] is
\[y - 0 = \frac{\frac{84}{23} - 0}{\frac{- 35}{23} - 0}\left( x - 0 \right)\]
\[ \Rightarrow y = \frac{84}{- 35}x\]
\[ \Rightarrow y = - \frac{12}{5}x\]
\[ \Rightarrow 12x + 5y = 0\]
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