Question

The equation of a line through \[(3, - 4)\]and perpendicular to the line \[3x + 4y = 5\] is
 A) \[4x + 3y = 24\;\]
B) \[y - 4 = (x + 3)\]
C) \[\;3y - 4x = 24\;\;\]
D) \[\;\;y + 4 = 43(x - 3)\]

Answer 1

Hint: Straight line is a set of infinites points in which all points are linear. Slope of the required line is calculated by using the property of product of slope of two perpendicular lines. Product of slope of two perpendicular lines is \[ - 1\]. In order to find the equation of required line use slope and point through which line passes.



Formula Used:\({m_1}.{m_2} = - 1\)
Where
\({m_1}\)is slope of given straight line
\({m_2}\) is a slope of required straight line
Equation of required line :
\[y - {y_1} = m\left( {x - {x_1}} \right)\]
Where
m is slope



Complete step by step solution:Given: coordinate of point through which line passes and required line is perpendicular to line \[3x + 4y = 5\]
Now given line can be written as
\[y = \dfrac{{ - 3}}{4}x + \dfrac{5}{4}\]


Slope of this line =\({m_1}\)
\[{m_1} = \dfrac{{ - 3}}{4}\]
We know that
\({m_1}.{m_2} = - 1\)
\({m_2}\) is a slope of required straight line
\[{m_2} = \dfrac{4}{3}\]
Required line is passes through point \[(3, - 4)\] so this coordinates must satisfy the equation of given
Equation of required line:
\[y - {y_1} = m\left( {x - {x_1}} \right)\]
\[(y + 4) = \dfrac{4}{3}(x - 3).\]



Option ‘D’ is correct

Note: Product of slope of two perpendicular line is\[ - 1\]. Line which are parallel to each other are having equal slope. Here equation of line to which required line are perpendicular is given. So use property of product of slope of two perpendicular lines. Every line makes some angle with axes. Slope of a line is also equal to the tan of angle which lines make with axes. Slope is also known as gradient.