Question

The equation of a line passing through the point of intersection of the lines \[x + 5y + 7 = 0\], \[3x + 2y - 5 = 0\], and perpendicular to the line \[7x + 2y - 5 = 0\], is given by
A. \[2x-7y-20 = 0\]
B. \[2x + 7y-20 = 0\]
C. \[-2x + 7y-20 = 0\]
D. \[2x + 7y + 20 = 0\]

Answer 1

Hint: On solving the two given equations we will obtain the point of intersection. From the equation of perpendicular given we can obtain the slope. The relationship between the slope of two perpendicular lines is \[{m_1}.{m_2} = - 1\]. Hence, we can find the slope of the equation of line. By substituting the point of intersection and the slope obtained in \[\left( {y{\text{ }}-{\text{ }}{y_1}} \right) = m\left( {x{\text{ }}-{\text{ }}{x_1}} \right)\] , we can obtain the equation of line.

Complete step by step solution: 
The line passes through the point of intersection of the lines whose equation is given as,
\[x + 5y + 7 = 0{\text{ }} \ldots \left( 1 \right)\]
\[3x + 2y-5 = 0 \ldots \left( 2 \right)\]
 On Multiplying \[(1)\] by 3 and subtracting \[(2)\] from \[(1)\]we get,
\[13y = - 26\]
\[ = > {\text{ }}y = - 2\]
Substituting the value of y in equation \[(1)\] we get,
\[x + 5( - 2) + 7 = 0{\text{ }}\]
\[ = > {\text{ }}x = 3\]
Therefore, the point of intersection of the lines is \[\left( {3, - 2} \right)\]
We are given that the equation of line is \[7x + 2y-5 = 0\].
The slope \[{m_1} = \dfrac{{ - a}}{b} = \dfrac{{ - 7}}{2}\]
If the slopes of the two perpendicular lines are \[{m_1}\], \[{m_2}\], then we can represent the relationship between the slope of perpendicular lines with the formula \[{m_1}.{m_2} = - 1\].
Therefore, Slope of perpendicular line is, \[{m_2}\; = \dfrac{2}{7}\] (since \[{m_1}.{m_2} = - 1\])
Equation of line passing through \[\left( {3, - 2} \right)\]with slope \[\dfrac{2}{7}\]is
\[\left( {y{\text{ }}-{\text{ }}{y_1}} \right) = m\left( {x{\text{ }}-{\text{ }}{x_1}} \right)\]
\[ = > {\text{ }}\left( {y + 2} \right) = \left( {\dfrac{2}{7}} \right)\left( {x-3} \right)\]
\[ = > {\text{ }}7y + {\text{1}}4 = 2x-6\]
\[ = > {\text{ }}2x-7y-20 = 0\]
Hence the correct option is A.

Note: The relationship between the slope of two perpendicular lines is \[{m_1}.{m_2} = - 1\] . The "point-slope" form of the equation of a straight line is \[y - {y_1}\; = m{\text{ }}\left( {x - {\text{ }}{x_1}} \right)\]. This can be used when we know one point on the line i.e., \[\left( {{x_1},{y_1}} \right)\] and the slope of the line.