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Question
The equation of the line with slope −3/2 and which is concurrent with the lines 4x + 3y − 7 = 0 and 8x + 5y − 1 = 0 is
Options
3x + 2y − 63 = 0
3x + 2y − 2 = 0
2y − 3x − 2 = 0
none of these
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Solution
3x + 2y − 2 = 0
Given:
4x + 3y − 7 = 0 ... (1)
8x + 5y − 1 = 0 ... (2)
The equation of the line with slope \[- \frac{3}{2}\] is given below: \[y = - \frac{3}{2}x + c\] \[\Rightarrow \frac{3}{2}x + y - c = 0\] ... (3)
The lines (1), (2) and (3) are concurrent.
\[\therefore \begin{vmatrix}4 & 3 & - 7 \\ 8 & 5 & - 1 \\ \frac{3}{2} & 1 & - c\end{vmatrix} = 0\]
\[ \Rightarrow 4\left( - 5c + 1 \right) - 3\left( - 8c + \frac{3}{2} \right) - 7\left( 8 - \frac{15}{2} \right) = 0\]
\[ \Rightarrow - 20c + 4 + 24c - \frac{9}{2} - 56 + \frac{105}{2} = 0\]
\[ \Rightarrow \frac{- 40c + 8 + 48c - 9 - 112 + 105}{2} = 0\]
\[ \Rightarrow 8c = 8\]
\[ \Rightarrow c = 1\]
On substituting c = 1 in \[y = - \frac{3}{2}x + c\], we get:
\[y = - \frac{3}{2}x + 1\]
\[ \Rightarrow 3x + 2y - 2 = 0\]
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