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Question
Find the equation of the perpendicular to the line segment joining (4, 3) and (−1, 1) if it cuts off an intercept −3 from y-axis.
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Solution
Let m be the slope of the required line.
Here, c = y-intercept = \[-\] 3
Slope of the line joining the points (4, 3) and (−1, 1) = \[\frac{1 - 3}{- 1 - 4} = \frac{2}{5}\]
It is given that the required line is perpendicular to the line joining the points (4, 3) and (−1, 1).
\[\therefore m \times \text { Slope of the line joining the points } \left( 4, 3 \right) and \left( - 1, 1 \right) = - 1\]
\[ \Rightarrow m \times \frac{2}{5} = - 1\]
\[ \Rightarrow m = \frac{- 5}{2}\]
Substituting the values of m and c in y = mx + c, we get:
\[y = - \frac{5}{2}x - 3 \]
\[ \Rightarrow 5 x + 2y + 6 = 0\]
Hence, the equation of the required line is 5x + 2y + 6 = 0.
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