Question

Find the equation of the line which is perpendicular to $ 3x + 4y + 6 = 0 $ and makes $ 0 $ intercepts with both the axes. Choose the correct option below.
1) $ y = 4 $
2) $ 4x - 3y + 8 = 0 $
3) $ 4x - 3y = 0 $
4) $ 4x - 3y + 6 = 0 $

Answer 1

Hint: We can solve this problem by using graphs. Or we can solve this problem by using some techniques which are by eliminating options. In both ways, we can get the correct option. But how to solve this question is determining the slope and a point passing through it.

Complete step-by-step answer:
Given that, the required line has zero intercepts on both X-axis, Y-axis.
This means the line passes through the origin.
Now we got a point that the required line passes through that is $ \left( {0,0} \right) $ .
Now we need to get the slope to determine the equation of the required line.
We have given that the required line is perpendicular to the line $ 3x + 4y + 6 = 0 $ .
We know that the slope of a line whose equation of the line is $ ax + by + c = 0 $ is
$\dfrac{{ - a}}{b} $ .
So, the slope of the given line equation $ 3x + 4y + 6 = 0 $ is $ \dfrac{{ - 3}}{4} $
Let us assume the slope of the required line is $ m $ .
We know that the slopes of the perpendicular lines are multiplied to get $ - 1 $ .
That means if $ {m_1},{m_2} $ are the slopes of perpendicular lines then,
 $ {m_1}.{m_2} = - 1 $ ,
Let us substitute the slopes of the given line and the required line since they are perpendicular.
 $ \Rightarrow m.\left( { - \dfrac{3}{4}} \right) = - 1 $
After simplification, we get
 $ \Rightarrow m = \dfrac{4}{3} $ .
So the slope of the required line is $ \dfrac{4}{3} $ and it passes through $ \left( {0,0} \right) $
We know that the line equation of the line which passes through $ \left( {{x_1},{y_1}} \right) $ and has slope $ m $ is $ y - {y_1} = m\left( {x - {x_1}} \right) $
So, the line equation of the required line is $ y - 0 = \dfrac{4}{3}\left( {x - 0} \right) $ ,
After simplification, we get
 $ \Rightarrow 4x - 3y = 0 $ .
Therefore the line equation of the required line is $ 4x - 3y = 0 $ .
The correct option is 3.
So, the correct answer is “Option 3”.

Note: This is a problem that can save time in the exam. We can determine the answer by looking at the options and eliminating them. We have given that the required line has zero intercepts on both axes. There is only one option in the given options which has zero intercepts on both the axes. By using this process we can save time in the examination.